Keeping it dry is the important part, ensuring it is dry before
you store it is just as important. I keep all of my gear (except my PolarPod) in my shop, in large totes with well fitting lids. I keep the PP in a noseeum bag the size of a king pillowcase in my closet, but have kept it in a tote outside in the shop while remodeling my bedroom. All of our gear, including the down products and the very large bags of down are in a non climate controlled shop, but the shop is dry and the products are kept in some type of container.
I may have misread what you meant but why are you storing for the winter? The winter is one of the best seasons for a hammock, IMHO