Fridge0 does not need a big battery bank, since it stores cold in its thermal mass and runs only when solar power is available.
But, at least a mimimal battery is needed to run the computer control. Probably on the order of 5 watts for a computer like a raspberry pi; this could be reduced more with a more embedded computer like an arduino.
Enough battery to run the fridge for a minute or two is also needed. Consider what happens when the fridge is running, and the sun goes behind a rain cloud. Suddenly, there's not enough power to run it, but it's still turned on. The inverter will try to run with whatever small solar power is still coming in, but it's not enough. Some inverters might manage a clean power off, probably accompanied with loud beeping. Other inverters might fail in more interesting ways.
To deal with this situation, the computer needs to check the incoming solar power frequently, and power off the fridge if there's not enough. The battery is needed to keep the inverter and fridge running until that happens.
A supercapacitor might also work, instead of a battery, since the total power that needs to be stored is relatively small.
Joey Hess's fridge0
My house has a battery bank used for other things than fridge0 (lights, laptops, internet), so I use that. I didn't need to increase the battery bank any for fridge0. It consists of 4 deep cycle golf cart batteries (model #SLIGC110), 860AH capacity. Total cost was ~$500.
I'm pretty sure all my fridge0 needs is a single small deep cycle battery, around 100-150 AH capacity.
My control software for fridge0 prioritizes charging the battery bank over running the fridge, unless the fridge is getting too warm. This way battery power is available for my other needs. Balancing these needs was one of the harder parts of the project; it involves understanding the state of charge of the battery and how much potential solar power is available.
others
Please add your information here if you do a fridge0 build.