I use those large plastic bins on wheels, as they stack, and are easy to use, and check them out regularly as I use up my stash. Fleece gets wrapped in old pillowcases before going in. I do try to be organised and use everything in order of being purchased, but you know how it is....... For my handspun, I use those pop-up laundry baskets as I can put them away when not in use, but at the moment they are full to overflowing, I don't know why!
I have two wheels, a mahogany saxony wheel thats actually not that large (my house is small) and a Babe upright charkha that comes to pieces and lives in a tote bag when not in use so storage is not a problem.
It is possible here in Adelaide to leave things in a shed as long as its well ventilated and the floor doesn't get damp. I stored my floor loom that way for several years, along with the rug yarn I had acquired with it. Sadly once I had the space to set it up, I no longer had the health to use it, so it is no more.
But........I now have a rigid heddle loom and have just found a book on Ebay that tells me how to use 3 RHs together to do overshot etc, so I'm looking forward to that arriving soon! Yayy!