Using Super Capacitors with solar cells and batteries to help save my dogs
You are going at this backwards. Always, always, always start by defining your requirements. Only then can you move on to considering solutions.

In this case, you need to first build a prototype dog feeding station. Only then can you get a feel for how much power/energy you'll need. With a nod to your idea of using supercaps, you can use motors which run at 5 volts. With a mechanism in hand, you can measure how much current the station draws (this establishes power requirements), and how long the motors run . From this you can determine how much energy you need. Remember to specify a worst-case power outage duration - if you size the system for 2 days but the grid is down for a week, your pups will not be happy. You'll want to run the feeder while you are home, anyways, in order to monitor it for reliability. After all, you don't want to find out that there's a weakness in the system by coming home to hungry dogs. 

Only now can you start looking at candidates for energy storage. I suspect (very strongly indeed) that student is correct, and supercaps are not the way to go - they simply don't store enough energy to run motors for any length of time.

I'd also cast a jaundiced eye on solar power for your backup. If you are absent when a storm blows through, and the sitters can't reach your place, how will you depend on the solar cells not taking damage during the blow?

Frankly, I'd be inclined to start out assuming a biggish battery backup charged from the power grid would be the default position for your needs.