Htydjtk

I've got a relatively new bike tire (hard case Bontrager) that got punctured by a nail. So now the tube bulges out of the tire and is of great risk of bursting (though I biked several miles before realizing this).

I'm wondering if it is possible to patch the tire? If so, how would you do this?

I'm guessing you'd want a non-stretchable patch.

To get you home, you can use a tyre boot. This is essentially a non stretchy patch that sticks on the inside of the tyre. They're not recommended for long term use but people do run them long term (for example relegating them to a commuting bike/wheelset). If you do use them long term it's a good idea to check then occasionally. This assumes tubed tyres.

Boots also don't work next to the bead as there isn't enough material for them to stick to and resist the considerable load.

I've improvised a boot from a glueless patch (and run the tyre soft). Crisp (chips) packets, gel wrappers and strong banknotes are also used. Something sticky to hold it in place while you reassemble and inflate is a good idea.

now the tube bulges out of the tire

It's likely time to get another new tire - this one is probably dead.

The tube bulging out of the tire means that the structural integrity of your tire is compromised - one of the functions a tire must do for you to ride safely is hold the tube securely. This tire no longer does that.

And a patch really can't fix it. The fibers embedded in the rubber compound are likely cut. It's those fibers that give the tire the strength it needs to hold the tube in place against the pressure of the air inside the tube, and no patch can uncut those fibers. A glued-on patch can fail, as the glue is not really designed to take the shearing forces that are pulling the tire apart at the cut. The tire's fibers are designed to withstand those forces, but they're cut.

If you do the math, the total forces put on the rim and tire by the air inside the tube are pretty substantial. For example, a 1.5" 29er tire has an area over 410 square inches (area of a torus). If that tire is inflated to mere 50 psi, that's a total of over 20,000 lb of force spread out over the area of the tire trying to pull it apart and push it off the rim.

A tire boot (as mentioned in what's currently the other answer) is, IMO, a temporary fix for a cut tire. It gets you home.

Tires are wear items - they wear out, sometimes faster than you'd like.

I did use a tube patch on a sidewall hole in the tire (I too had a nail) - and the tire bulging has decreased somewhat. It's not a long term solution though.

If the hole is not on the sidewall but on the tread, you might possibly solve it with an auto tire patch (the kind used for holes in the tread).

Keep in mind that auto tire patch (the one in the thread) is NOT TO BE USED for sidewall holes (once a car tire has a sidewall hole, you must buy a new one).