Maybe add some concrete blocks and encase the area so that rodents will not be able to easily enter in there. The floor can still remain dirt, or you could use some patio stones as a firmer floor with sand as a grout between the stones or ceent slabs. Also, a door of some kind, or at least an enclosed way to enter the area so that wildlife cannot go in to nest, roost, or make that their home too (like big raccoons!).

I think a lot of roots do well if stored in cool sand. You could put big plastic containers of sand in there and put the lids on to help keep things out. The sand was mentioned in the "store apple with potatoes to keep from sprouting" thread and I've also seen Alton Brown do it with carrots.

Melissa