Would it be possible to add banks of large capacitors inside the case of a conventional PC to act as a sort of short-term UPS to hold up the various supply lines (ie 3.3 volt, 5 and 12 volt, etc) for perhaps several seconds to maybe a minute or so just to overcome the temporary loss or a glitch in the mains AC power?
It would be possible but impractical, and your PC case would be the size of one wing of the Cray 1. That's because you want to extract most of the energy stored in the capacitors, and that means discharging them quite a bit - way outside of the tolerances of those supplies.
The only practical way of doing it with capacitors is by adding them to the high-voltage DC link of the ATX power supply, with proper safety and design precautions of course.
ATX power supplies are switching and typically will operate from 100-240V. So, if you're on 240VAC mains, the capacitors on the DC link can discharge from about 340V to 140V while the supply maintains good output voltages. This would extract >80% of the energy out of the capacitors. If you only got 120VAC mains, you'd need to step it up using an external transformer first or add a voltage doubler between the rectifier and the DC link.
As for how much capacitance you'd need? Say a 100W load, 100J per second of energy, 80% energy discharged from 340V - that's 2mF per second of hold-up time per each 100W of load. If you're gaming at 250W (let's say), you need 5mF per second of hold-up time. So, for example, 50,000uF for a 10s hold-up, and those have to be 450V capacitors. Not entirely impractical. Just remember that a 75% extra-fast stick of dynamite releases about 1MJ. 250W for 10s is 2.5kJ, so 1/4% of a stick of dynamite in energy terms. If you don't protect those capacitors and their circuit against short circuits, it'll literally blow in your face upon an inevitable short circuit. Each capacitor needs its own fuse.