The brains of fridge0 is what really makes it possible. It needs to determine when there's enough solar power to run the fridge, and turn it on, powering it off when power is lost. And it should avoid freezing the food in the fridge, so it needs to sense the temperature inside it as well.
Warning when the temperature goes outside of the safe temperature range is also a good idea.
This could be done in many many ways; most will probably involve a small embedded computer such as a raspberry pi, that runs all the time, a relay that the computer users to control power to the fridge, a waterproof temperature probe or two, and an interface to the charge controller to keep track of the current solar power situation.
Hardware enclosure has more on the hardware side of this.
Joey Hess's fridge0
I use a cubietruck computer which uses around 5 watts but is also my house's router and server.
I've had good luck with waterproof DS18B20 temperature sensors, which use the 1-wire bus. This one from Sparkfun is good (I created http://joeyh.name/blog/entry/easy-peasy-devicetree-squeezy/ to get the cubietruck to enable the 1-wire bus.)
I have one temperature probe in the air near the top of the fridge, so I measure the highest temperature inside it, and another inside the thermal mass of water.
For the relay, I use this kit from Sparkfun, which switches the AC power from the inverter to the fridge on and off.
The computer talks to my charge controller using the software at https://github.com/kasbert/epsolar-tracer
The software to integrate all this together in fridge0 is in my git repository at https://git.joeyh.name/index.cgi/joey/homepower.git/. As well as controlling fridge0 it generates graphs and also automates other parts of my house. If you want to adapt it to your build, I can probably help out. It uses haskell and FRP, as explained in https://joeyh.name/blog/entry/my_haskell_controlled_offgrid_fridge/
others
Please add your setup here if you do a fridge0 build.