This is not possible within your constraints

As others have mentioned, the main benefit modern firearms have over melee weapons is accuracy, fire rate and range. Thus, under your constraints, a charging swordsman will be hit multiple times by a gunman before he closes the gap. We must either have a shorter gap (to reduce the number of shots that can be fired), inaccurate or slow gunfire (to reduce the number of shots that can be fired/hit) or we must be able to shrug off bullets (due to armour technology outpacing firearms).

Due to your constraints, we cannot limit firearms via technology (such as by restricting the quality of raw materials or the quality of manufacturing) as doing so will mean your firearms are no longer "Modern".

By limiting the ability of both combatants to realistic levels, the swordsman will not be able to parry or dodge the bullets, and thus will get hit multiple times whilst approaching.

Your blanket term for Exceptional Circumstances appears to block all other options. Geographically restricting range is not allowed. Allowing the Swordsman to shrug off the bullets thanks to advanced armour/shielding is not allowed. Restricting the velocity of firearms due to sufficiently-advanced alien tech is definitely not allowed. Simply put, as your question stands, your constraints are too tight to permit a change that will bring melee weapons on par with firearms.

Every IRL National Military will want to use the best tools for the job after all. With your constraints basically being "Real-World", if the armies of the world can't find a reason to use melee weapons, it'll be because the reason does not exist.

Guns are by definition ranged weapons. To make melee preferable or comparable to ranged combat you need to remove the advantage that it has, the simplest way to do that is to remove range. Set combat in an environment so complex and obstructed that range is minimal.

Consider a rotating space station as an example, unless you have a central open space, the curve of the floors and low ceilings mean you can't actually see very far ahead of you. Add corridors, spiral stairs, suites of rooms and you're all but in someone's face before you have line of sight to them.

The same can be true of a slum situation, buildings so tight and spaces so narrow that unless you're following a bulldozer you don't have line of sight more than a few metres ahead of you.

Bullets rebound. This adds a level of risk to using firearms, that you don't get in the open field situations for which they're designed. Moving through a standard dungeon crawl cavern which just happens to be through a type of rock with a greater coefficient of restitution for whatever handwavium reason you choose. It's now dangerous to all parties to miss your primary target with a firearm as the rebounding bullet could hit anyone.

I'll try my hand at your comment on L.Dutch's Answer:

Very insightful. Giving a "wavehandium" reason for slow fire rate is feasible. However, is there a way to make modern firearms less accurate - maybe something about the world's physics?

Since you're writing a science fantasy setting, you could either play with the actual world (as in planet)'s conditions if the game is set on only one planet - or with the rules of physics themselves if you want it applicable on different planets too.

Planetary conditions

Let's try the more restricted one first: if we leave the laws of physics alone and only adjust conditions on one planet, what could be done to take guns' advantage away?

Bad visibility

You don't need slums' cramped conditions taking away sight line to the target if you can't see very far in the first place. A planet where fog, sandstorms, pollen, spores or swarms of insects are common enough that a gunman can't reliably expect to see the target in time to shoot them before they're in his face with a sword will be a planet where a gun isn't much better than said sword.

Bad conditions for technology

Several of the options above also work here.

A well-maintained gun shoots accurately and reliably. A gun with grains of sand (or crawling insects...) in the barrel might not shoot quite straight, especially after the first couple dozen shots when the barrel gets more and more worn from the effect of a bullet travelling through it and dragging sand along its length. The firing mechanism might be jammed by sand or rusted from the constant mist, the gunpowder might be damp... depending on the technology used, environmental conditions might make "will it shoot or not, and if it does, will the bullet fly straight?" a gamble.

Note that this becomes more and more manageable the higher your tech level and the more you're willing to pay - modern firearms can be built to fire reliably after taking them for a swim, but you're paying more for that than you would for a gun that only needs to work on the shooting range. Kalashnikovs have also anecdotically been known to work fine after being buried in dirt, so you'll have to restrict your world's engineering capabilities a bit if you want this to be a factor at more than the low-price range. If you're fine with plebs' guns being unreliable enough to counter a shooter with a kitchen knife but elite mercenaries firing reliable guns at elite swordsmen dodging matrix-style, this might be the way to go.

aggressive animals

Guns are loud. The Sky-Snakes will zero in on anyone firing a gun repeatedly, and those things are nasty and poisonous. The boombeetle will look for a mate where it hears the mating call, which sounds kinda like a gun being fired. Think of something fun ;)

missing is dangerous

See those rocks over there? Hit them too hard, like with a bullet or such, and they'll explode - shards everywhere, including your face.

See that plant over there? Those thick leaves are filled with a flammable, sticky liquid. Basically napalm. You do not want that splashing around. And that one has swamp gas in its fruit, explodey. The seeds are also poisonous.

Obviously this would lead to a world with no (or only the dumbest or most reckless) plebs walking around with guns, so it might not work for you.

Different laws of physics

If you're willing to meddle with the laws of physics, the sky is the limit... or rather, not even that. You could just handwave that it's impossible to manufacture explosives that are reliable enough for our kind of modern firearms because chemistry works differently, or maybe inertia follows a different law where things in motion don't stay in motion - they want to return to a state of rest, so bullets lose power quickly the further they travel. Maybe there's a material that acts non-newtonian in a way that makes it comfortable to wear but lets it stiffen quickly when hit with a rapid-moving bullet - but somehow this material is still vulnerable to cutting edges on spears, axes and the like...

Different Biology

Bonus: If your people don't have to be human, you could just give them quickly-sealing blood vessels and more redundant bodyparts. If a bullet hits you and just gets stuck in your dense muscle mass, or maybe pierces an artery but that artery seals up and the blood for the arm you've been hit in gets routed through the secondary set of arteries instead, or the bullet pierces just one of your three hearts - maybe you can keep fighting and recover from it. And maybe that's easier than recovering from having a piece cut off with a sword.

Fire rate and precision are the key factor for guns to be dominant over melee weapons.

Think of the bow: archers were used for quite a while after guns introduction simply because the fire rate they could grant was higher than the first guns and rifles, with a significant better precision.

Same here, if you fire a shot that goes within a large area around your pointing target and then have to wait a long time to reload the weapon you are basically giving an advantage to your enemy approaching you with a sword in his hands.

So, keep your guns primitive (heavy, slow firing, poor precision), and that would explain why melee weapons have an advantage.

Expense, make one of the prime bullet ingredients expensive or something like that and a sword is better since your gun turns into just a clumsy club without bullets. Some places still use swords for this reason. They just don't have much access to guns and bullets. They're not something you can easily make.

Even a pipe gun which can be made with some technical know how and tools cannot do much without bullets.

You can't without dropping at least one of your conditions. Considering that it's for a game, it's not as bad as it may sound.

I suggest you drop "exceptional" condition and introduce plethora of exceptional circumstances. Cramped slums, packed space stations (like Separatrix's answer), strangely bullet absorbing melee space monsters (for those you definitely want so called boar-spear with cross bar, to stab monster and keep it away from yourself at the same time), realistic sewers (instead of absurdly spacious ones) and so on. Since weapon usefulness is function of circumstances: guns are all around better, but action of the game happens in those exceptional locations and situations, focusing on people specialising in dealing with those exceptional situations. As a bonus, you can always have curveball wide open location/scenario with gun totting human enemies requiring either change of gear or complete change of tactics.

Reason why guns are superior is their range and accuracy. Times of volley fire making up for inaccuracy and slow reload are gone since mid 19th century and you can not reintroduce those in modern setting without breaking "exceptional" requirement. Dune uses shields, but there are other options, for example: short range teleport (usually called "blink" in fiction) removes range advantage, stealth (plain old hiding or sci-fi active camouflage) removes all advantages and let's call it "distortion shield" which doesn't stop bullets but turns anything short of exact hit to centre of mass into miss would remove accuracy advantage. Some of those break into "raw-ability" like territory, shooter with stealth (basically a a sniper) is better than swordsman with stealth, blink goes both ways, with shooter being able to escape (potentially counter-able by another teleport by melee fighter) while distortion wouldn't be very effective against melee weapons giving shooter advantage only against other shooters.

Speed control.

In this season of The Expanse, there is some alien tech which limits the speed of any object (relative, I think, to this central artifact). If the object exceeds the limit it stops.

This would decrease the killing power of guns because kinetic energy is mv^2; a bullet could knock you down or break a bone but not produce supersonic shockwaves in a body. You could cap bullet power wherever you see fit.

Make the gravity of the planet higher.

Quite simply, this has the effects of increasing the atmospheric pressure, leading to higher turbulence and reduced bullet travel distance, plus the increase in gravity means the average person has to be stronger, thereby making a sword able to penetrate more armors. In addition, the higher gravity means the bullet drops further faster. This doesn't affect accuracy though, only the require skill to use a gun and the maximum range of a gun.

The other fun effects of this could range, from having more dense people (Bullets do less damage) to reducing atmospheric visibility to such an extent that you can't target people with a gun at substantial range anymore. People might also develop stronger armors, due to the need for stronger materials in a world where everything is heavier, and be able to wear heavier armors, due to the inherent strength increase of a person who grows on a larger world. (Which can be assumed due to the increased gravity)

However, the most obvious way to demonstrate this would be using a brick falling from height. Now, instead of falling 5m in two seconds, it might fall 50m in the same time, leading to a bullet that's in the air for potentially .1 seconds to be hitting the dirt within roughly 30m. (assumed from the speed of some shotguns of 1000ft/second)

If with these advantages swords are still unusable, remember that the effective range of a gun by your average soldier would be ~5 meters, due to the extreme bullet drop and the increased turbulence from a denser atmosphere making hitting a shot hard. A thrown weapon would be unaffected at short ranges, which works well for this because even now spears aren't effective at much longer ranges.

Superior ballistic armor

I'm not sure whether this counts against your exceptional circumstances (as assumed by Kyyshak), but it seems the least exceptional way to eliminate the overwhelming advantages of modern firearms over melee weapons among well-equipped combatants.

On your world master armorers have discovered a (very expensive) way to make armor that stops most bullets outright, except at point blank range or from certain angles. It doesn't fare quite as well against slower weapons, especially slashing types like swords.
A skilled combatant can pierce the armor with a gun after a bunch of tries, putting the gun on a somewhat equal footing with a sword.

Defending against a gun while using a sword doesn't mean dodging bullets, but rather preventing the opponent from aiming at and getting a shot on weak points in the armor. This doesn't involve Jedi foresight or acrobatics, but an aggressive style that puts the sword wherever the gunwielder's arm is or would need to be for a shot. As soon as the gunwielder backs away, the sword user can angle their body to minimize exposure, making the gun mostly useless until the distance is closed again.

Against anyone that doesn't have the resources (and permit?) to own a set of armor, firearms are the weapon of choice. So the plebs would be shooting eachother up like normal, as you want to.

Honor

Your society might have a very particular view of ranged vs melee.
Over many generations people have been indoctrinated to believe that there is more honor is fighting melee versus using a ranged weapon.

Most plebs who's honor is not worth mentioning would resort to ranged weapons while the accomplished mercenary would wield a sword showing the rest of the world that they don't even need a gun.

A mercenary who only uses melee weapons to complete his contracts would be more famous and sought after then a mercenary who just uses a 10 mile ranged sniper rifle. Even if the sniper rifle is technically a better weapon it would only be used by those of disrepute.

TLDR - The culture puts a great value on being able to take an enemy out at close range. People strive to get good at melee and see ranged as a handicap.

Better advances in protection.

Currently ballistic firearms have progressed faster than ballistic protection, which allows guns to be so dangerous. But gun ballistics progression is nearing it's end unless we transcend towards railguns and other exotic weapons, while metamaterials are becoming more and more attainable.

Imagine spidersilk armors with Graphene layers and other lightweight, extreme performance materials. If these become cheaply available the guns required to kill your opponent will have to go big, and even a. 50cal could start having trouble killing the opponent and would rather break bones than kill.

This in turn allows melee weapons to become viable again. In Iraq and Afghanistan it's estimated that the trained army outnumbering their opponents still needed 250.000 bullets per insurgent. If you need bigger weapons and more hits to down a person it becomes less interesting to use that gun. Big weapons like a. 50cal have lower rates of fire (when handheld), are unwieldly and very hard to aim unless mounted or resting on a surface. Your opponent has to grab a smaller caliber and hit you multiple times to bruise and break you enough to knock you out of the fight. Time you can spend rushing your opponent and smacking him with a mace or similar. Even the good old knights of old in slash-resistant armor were very vulnerable to getting bashed.

Add in potential upgrades in your melee weapons unavailable to bullets like high voltage, potential superheated edges, monomolecular edges etc and you can make melee weapons more useful.

Here's a simple idea:

Much higher atmospheric density leads to several disadvantages to guns: 1) higher required charge to reach decent muzzle velocity & faster velocity drop-off in air - hence steep trajectory curves; 2) soundwave propagation in denser atmosphere means concussion from said shot would be loud enough to damage shooter and be highly detectable no matter what; leading to 3) the edged weapons are just inherently stealthier - you get close, katana, gim or tonto undrawn, once inside striking range you execute an aido-style quick draw, and resheathe removing the blood, as your silently halved opponent drops into separate pieces.

Most properly we can say that drag forces scale with the square of velocity and with the density of the fluid - see equation below, where p is media density.

All you need are reasons rendering things more even.

A few ideas:

Movement
Sufficient mobility makes smaller weapons superior. If you can clear the distance to a guy with a gun before he can draw a bead, melee is superior. (Slowing guns down could work similarly, but that raises other questions. Still, if all ballistics slowed but retained momentum, if not all the impulse, you'd still have dangerous weapons, albeit less useful.)

Invisibility or major resistance to damage while moving is roughly equivalent to rapid movement for gun vs. blade purposes.

As to how to to get this high speed or safe movement, well, that could be a common device or spell or even a martial technique if it's not a normal fact of the world. (Heavy mists could render things invisible out to a few hundred feet and blurry. Basically, a PS1 game world...) Which leads us to...

Chi
It's not realistic, but of your inspirations, only one even vaguely hints at such. Special martial training can unlock abilities that rival the range of guns but without some of the drawbacks. Guns could have the advantage of being powerful without the chi requirements. The disadvantage being that chi isn't a big benefit to firearms, limiting the upper end of lethality.

This gets you the common schlub using guns with warriors using other weapons thing. (This could be a universal trait of the world rather than a special thingy. Depending on how you interpret things, that could avoid your special power thing.)

Shielding
The shields in Dune could be tweaked to something that could explain this. A narrow impulse is easily deflected while a cut can drop the whole shield, say. This needn't be a technological marvel, either, but could be a magical or field thing that effects people for some reason.

Slow bullets plus the ability to deflect them with a weapon could lead to safer swordsman, a la Force blaster deflecting tricks but nigh universally acheivable.