Water comes into shed

The previous owners of our house built a shed on a concrete slab (Diablo iii Gold). Shed is about 8' x 10', 2x4s and pressed-wood-fiber siding, and isn't sealed to the slab. The slab has enough of a slope that when it rains, water runs into the shed.

After losing some stuff to water damage, I temporarily laid down 2x4s with plastic on top, so even if the 2x4s get wet, it wouldn't wick to the boxes stored on them. This doesn't seem the best solution, though. The boards could mildew, and/or mice could set up housekeeping in the space. (I found mice in the shed last winter, and have been sealing the access points as I find them.)

I've thought of using plastic decking planks which wouldn't mildew, but they'd still create a gap. I thought of wrapping the system in hardware cloth to keep varmints out.

I considered spraying GreatStuff or other caulk into the gap between the shed and the slab (Diablo iii Gold), but the concrete would still get wet, and might mold or wick into the shed anyway.

I'm in Los Angeles, so frost isn't a problem.

Any other ideas? Any of my ideas sound better than i think?