If you don't want to (or haven't presently got the materials to) cobble together ventilated storage bins with mouseproof screening, here is a trick you might try. I use it to keep mice from eating the seedlings I start under lights in my basement.
I put the seedlings on a card table. Keep it away from walls (and preferably under a mouseproof part of the ceiling i.e. not exposed wires etc they can run atop). To prevent them running up the table legs, I took icecream tub lids (but you could use anything similar, just make sure it is at least 10" or more in diameter, -ish) and cut a hole in the center. Push the lid up the table leg. If the leg is round, push it til it binds (should be at least 2' off the floor); if it's square, push it to that height and then slather on the duct tape so mice can't squeeze through the gap. Duct-tape these 'mouse guards' firmly into place on each leg. Now they're like squirrel baffles on a bird-feeder! It has worked for me for 2 years now, REALLY well, whereas before I was losing like all of my seedlings.
Be aware that I think some of those veggies actually need cool and *damp* -- look up the preferred storage temps and humidities for each of them.
Good luck,
Pat
