Here's her actual submission [1]. She created a novel dielectric and achieved an energy density of 20.1 Wh/kg which is about half of that of a lead-acid battery, and something like 6%-10% of a Lithium Polymer battery. She lists a capacitance per volume, but nothing about breakdown voltages, so I can't say what the size of her capacitor would theoretically be for 1200F rated at 6V. Nor can I say if it'd even be feasible without knowing how difficult/expensive the dielectric is to manufacture.
Also, here's a page [2] with a list of common battery chemistries and ballpark figures for their energy density values (about halfway down in green). Newer Li Poly tech is somewhere in the 180Wh/kg range.
The sad thing about this is that we're missing a bigger point. Capacitors are useful because of their low internal impedance. They're able to dump (or receive) a tremendous amount of power incredibly quickly. It's useful when you have a circuit which needs a high-current burst which is powered from a low current (but high capacity) power source.
1: http://www.usc.edu/CSSF/Current/Projects/S0912.pdf
2: http://www.allaboutbatteries.com/Battery-Energy.html