Go buy the biggest Rubbermaid Trash cans on wheels. It will hold tons of fabric! It can sit right outside your back door, the lids fit very tight and have handle locks on them. Put a zip lock bag with slits cut in it full of charcoal in with the fabric to keep all the musty smells away.
My kids seemed to move all the time before settling down. We used these big trash cans to move every household items they owned. Easy and no bending over to pick up boxes. They roll right in a truck and right off. All of the ones filled with books were left outside and no rain ever got inside of them.