I believe I have seen storage units made of open weave wire covered in plastic. They would allow you to stack fabric etc. inside the units without it falling through unless it is very small and allow air to circulate also. The fabric would also still be visible which would save a lot of digging. You may have inspired me to try reorganizing my stash using such shelving. If actual shelving is not available perhaps you could just purchase a lot of the plastic milk carton like bins and stack them on their sides to create your own shelving unit.