Using a high C capacitor
You don't want a capacitor solution when you draw that large currents for such a long time. The word is battery. Why don't you use that? It's cheap and can store a lot more energy than a capacitor the same size.  

This capacitor is "only" 50 Farad and already costs 1500 dollar. How many batteries can you buy for that? It also stores the amazing amount of 8000 J! An typical AA NiMH will store 12000 J (1.2V \$\times\$ 2800mAh \$\times\$ 3600 seconds). Less than 3 dollar.  



One application where the use of high capacitance caps is justified is in a strobe flash. For my photography I have a set of studio strobe flashes rated at 500Ws. These 500J are dumped in 1/2000 second, which means that for a split second you have to supply 1MW. No battery can dump so much energy so fast.