Just thinking about this quickly...I don't think condensation will be a problem, or you could make it not a problem. Here's how/why:
1. If you give EVERYTHING in the computer a good cleaning and use surgical/latex gloves when assembling it, there should be no sweat/salt/electrolytes on your computer components. When you put it in the fridge, even if there is condensation on it, pure water is not a very good conductor. So it "should not" be a problem.
2. Put it in the freezer!

Even if you get condensation on it, ice is an even worse conductor! You could even put it inside an extra large Ziplock freezer bag

3. If the refrigerator is only for the computer, you shouldn't be opening and closing it much, so no condensation would work it's way in. If worried about it, then put a bunch of moisture absorbing material in it (think you can by solid, disposable moisture absorbent at most hardware stores).
Sound's like a great idea! Send pictures!
Edit: conductivity measurements...
pure water: 0.055 µS/cm ocean water: 53 mS/cm That's a factor of almost 100 million.
I think you're safe if everything is clean before you put it in.