I know that zero is a finite number, but since the photon is a massless particle, wouldn't it be impossible to get a kilogram worth of photons bouncing around in an optical cavity?

No, it’s entirely possible in principle. If you build a perfect optical cavity, weigh it, pump a bunch of light in, and weigh it again, you’ll find that the weight went up by ghf/c^2 summed over each photon added to the cavity. Divide by g and you get the sum of the masses of the photons. 1 kg of photons would be enough to destroy your lab and the rest of your city, so only a very advanced civilization would ever do the experiment on this scale.

Photons indeed have no rest mass, but it’s impossible to ever find a photon at rest unless you yourself have no rest mass, in which case you also have no ability to do experiments.

Actually, as a physicist with no rest mass, I'm perfectly capable of performing experiments. I just never seem to find the time...

Photons have zero rest mass but it so happens that they are always zipping around at the speed of light so their actual mass as observed is nonzero.

Only in practical terms, as that’s c^2 joules of photons, which is roughly equivalent to the Tsar Bomba detonation.

Something fun of a related nature; if you heat something up, zero particles added, it gets heavier :)

For some reason, zero is often considered not finite. I personally find this terminology quite unintuitive, but it is well established.