I have found plastic bins with covers to be fine for storing fabric long-term. The bad plastic for storing fabrics is soft plastics, as in plastic bags. These "outgas" chemicals and are not immune to mice, who can chew through the soft plastic. Hard plastic bins do not outgas and are pretty good about keeping mice out. Most allow a little bit of air exchange, which is good for fabric. Some people drill a few holes in the bins so air can circulate, but I have never done that. All of my fabric bins have been stored in the house, though. If stored in a non-temperature controlled unit for six months or more, it might be a good idea to drill very small holes in the lids -- so that if there is any condensation inside the bins from extreme changes in temperature, the moisture has a chance to escape. Moisture that cannot escape would cause mold in the fabric.