Earlier quoted context omitted.
Instead of 10 V and 2 F, how about going for 2.5 V and 50 F? A capacitor with those specs is only 40 mm long and 18 mm diameter [1]. That shouldn't be too hard to fit in a typical server or desktop. That's under $4 in quantity. [1] http://www.mouser.com/ds/2/257/Maxwell_HCSeries_DS_1013793-9...
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 curr…
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!)