I also use an old mini/wine-fridge, that I got for free at the end of an estate sale. My shelves are not so great, but I plan to make some out of hardwood. Other than that, it works great. The fridge does offer a bit of mystique and cache to my cheeses to guests who come over. They all want to know what kind of goodies I have in there ...LOL.
As for storing cheese in this type of, "cave," it really depends on what type of cheese you are keeping. If you have moldy cheeses such as camembert, or stilton, wrap them loosely in cheese wrap & store those each in a separate plastic Tupperware type of container, with the top slightly ajar if they get too moist. Hard cheeses can be slipped into a plastic baggy with a paper towel and left slightly open. Semi-soft cheeses place on a sushi mat will also help to keep air circulating around the cheese. The more cheese you have, the higher the humidity will be. If it is still too low, put a semi-covered bowl of water in the bottom of the fridge.