When we made our last move all our household things were in several of those rental storage units for over 6 months. I packed all my fabric in large blue tupperware storage bins. Inside the bins the fabric was in garbage bags. For the wool that I had in storage, I threw in some moth balls with it just to dissuade bugs from taking up residence. All the fabric was fine when I finally unpacked it.
lol - and 8 years later .... some of it is in the "cubbies" upstairs in the bins