How to insulate wood box from cold outside air flowing into house

I'm moving into a cute, little old one bedroom barn apartment in the country. It has a built-in wood box for storing firewood near the stove. The box is 24 high 38 long 21 across. It has a door that closes, however the door doesn't stop the outside cold air from coming in. The wood box construction isn't sealed, and made of plywood. I can see the ground below through cracks in the plywood. Is there any way I can insulate this entire box? I have a nice landlord who would probably gladly help me, but I don't want it to be expensive I don't even need to use the wood box. There's plenty of room to put some firewood pieces on the brick area by the stove. Should I get some sheets of fiberglass and lay that down and seal it off with plastic?