You have just described a UPS. Or a laptop. A battery just makes more sense than a capacitor in this application.
[1] http://www.mouser.com/ds/2/257/Maxwell_HCSeries_DS_1013793-9...
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!)