Fridges are not build to run 24/7 and the compressor will burn out rather quickly. Additionally as mentioned by dsenette, condensation may be an issue if you are in a humid environment, though most modern fridges remove the humidity from the air so it shouldn't be too much of an issue.
If you are serious about getting below ambient temps, the only method that you can try is cold plates, LN2 and TEC cooling, among a few others. Many of which are VERY expensive and in the case of LN2, very, very risky and should only be performed on hardware you're willing to lose.
Lastly, if you do try this, which I don't recommend, remember to remove your hard drive from the cold. Hard drives do not like a cold environment.
Edited by james_8970, 29 November 2008 - 09:38 PM.