Your approach is a good idea and bad idea... is it possible? Not with just the fridge alone... first calculate the amount of cubic space a fridge can cool... without a heat source inside... its impractical...
However just to kick up some condensation on this thread over 4 years old, here's a solution... first off you wouldn't do anything crazy like that to save money, but heres a way to do it for half the price of a custom water loop and is mildly economical!!!
Items needed - used gun display case (large glass 3 window and metal back), window silicon, duct tape, cheap atx tower, all your cpu parts and of course the cooling aparatus will be a 3 in 1 floor air conditioner with remote. also note you'll need other materials for a furnace duct system to direct air or acrylic sheeting. Get creative with it.
Why a gun display case? Its easy to seal with silicon...
The most important part aside from pc and case is the cooling apparatus. The 3 in 1 comes with a dehumidifier. You will want to run a drain hose out of the gun display case from it into a bucket at least. So why a 3 in 1 floor ac unit? Its all about the squarefootage baby! Lets say gun display is 3'x3'x6'=54 cubic feet. A 3 in 1 rated for 300 squarefeet, which rated for a standard 8' ceiling... so 10'x30'x8=2400 cubic feet to be cooled against high watt devices like tv's, fridges, deep freezer, gaming consoles, pc's (wink). It gets better! The 3 in 1 also comes with a large sp fan that is filtered!
So on your 3 in 1 you have 3 air ports. Heat exhaust, filtered intake (both on back) and cold exhaust on front. Luckily the 3 in 1 comes with a heat exhaust extension that installs in any house window.
Build it how ever you like but here is the technical break down. After installing everything, seal the door with tape and use the remote to power on the 3 in 1 in dehumidifier mode. Wait awhile and check humidity, once humidity lowers below 30 then its safe to switch to cool which will also dehumidify the air further. With the cooling exhaust blasting on the cut up tower to optimize air on it with it jammed inside a duct vent that is mounted and supported to cooling exhaust. Watch your temp probes and enjoy... you got a ton of space in that case to get really creative! Like turn a old acrylic aquarium into cpu case... i left out a lot of the creative stuff, but hey its your design, your money, your baby and you can control the humidity levels as well as temp via remote... but at least it will work! i also don't reccomend dropping the case temp to low, about 60 degree's inside case and 70 degree's inside room... you'll get condensation on outside and have to keep drying it unless you dehumidify the room the setup is in. To optimize it, install liquid cooled radiators for gpu's and cpu, just run the radiators behind the 3 in 1 filtered intake to cool heat. I would reccomend a insultator and custom mount plate. Be just like server cooled warehouse...
Also for a fridge build, let me know if you see any old cake display fridges like they have in old style diners that broken. i got crazy idea for a pc build involving one. note the fridge part is useless and bleep the cake, i got another party in mind. A mad scientist party!