Search digikey.com for "supercapacitor".  There you will find a number of capacitors listed in the 1200 F to 5000 F range.  At this moment they offer 1700 F @ 2.7 V for $160, 3500 F @ 2.7 V for $210, and 5000 F @ 2.7 V for $255. These are made by NessCap Co Ltd.  So they do appear to be available, though I would not say that this is an indication that they are widely available.