Theoretically it's possible, but in practice the lower input voltage makes it harder to convert it to +5/+12V. It becomes increasingly lossy and expensive - to convert 0.5V to 12V at 10W is not trivial to begin with. Voltage across a capacitor drops continuously while discharging (unlike a battery). So with a 2.5V capacitor, being able to use it between 1.25V and 2.5V is already pushing it. On average, discharge current at 10W is around 8A. The internal resistance of the capacitor (ESR) better be very low (it probably isn't) at this low voltage - even if it's 0.1 Ohm, at 8A we already lost 0.8V from out meager 2.5V, and now the useful energy is just 1.7-1.25 = 0.5C0.45V^2 = 5J, just enough for 500ms at 10W.

Interesting. 500 ms would would probably not be enough time to save everything (although maybe on an SSD based system it would be...), but it would probably be enough time to save select information that would allow ensuring that the disk is in a consistent state.

The 10 F capacitor has an ESR of 0.075 ohm, but that's at 1 A. They have a 100 F model that is 0.015 ohm at 10 A, and a 150 F that is 0.015 at 15 A. Based on your calculations, these look like they would have a good chance of giving enough time to save everything (especially on an SSD system).

Those are physically bigger but should still fit in a normal desktop or server.

(There is another manufacturer that has up to 630 F!)