No, it would work, but condensation is a problem. This issue has been discussed many times everywhere. Things will sweat, especially when you open the fridge they will sweat hella bad, but because of the drastic temp difference there will be lots of condensation and we all know that water and electronics dont mix.
You need to build yourself a phase change system. Once you do that, you have to properly insulate every peice in your computer. It is a costly, very involved process, but it proves to be excellent cooling.
Also, you really dont need to worry about temperature being too cold until around -200ºC. But you dont want to cool things like your capacitors to sub zero temps, they dont like to hold a charge when they are too cold. However, yoour CPU LOOOOOOVES being cold!!!