Hi
I can understand the lack of space to store sheets (or anything for that matter). It does say in the EYFS to provide enough clean bedding and I don;t think it would be well received if we asked our parents to provide or wash sheets. Our children have individual bags (without handles or drawstrings) for that weeks clean bedding which we keep in a box (like those ikea folding ones) and kept with the beds or on a floating shelf at the other smaller setting. the children use their pictures on these bags to find their own beds and then keep them underneath while they sleep.
Hope that helps, theres not many responses on this forum these days