Are you talking about a wooden ice box, like this: http://www.rockler.com/product.cfm?page=187?
I think it's a great idea in theory. It would look fantastic. You just need to insulate with rigid foam and put gaskets on all the doors. Using the guts of an old mini-fridge won't be easy, but I think it's doable. I would actually recommend making your own icebox from scratch to make this part easier. You can integrate the insulation and cooling as you build and assemble, rather than having to retrofit an existing icebox.
My only concern is the interior size. Would you use it for a beer fridge? Or are you going to try to stuff carboys in there? Not sure if you can fit more than one if going with the 3-door design. You could always make the left side be one large door that looks like
two separate doors. Then you could probably size it to hold two carboys.
Also, the "Similar Threads" below look like they have some ideas you can consider.