I live in a small house and have had to store most of my fabric in a good-quality, weather-tight shed my husband built. It's been there about 5 years now. I should add that we are in New England, where winters are tough and summers are hot and sometimes humid. Some is in plastic bins and there has been no problem. Most of it is in cardboard Banker's boxes and a clean garbage bag surrounds that and is sealed tightly. They are on shelves. I have had no trouble with either category. I also have some donated fabric in open boxes on shelves and from time to time I find a dead bug, but that's about it. No light gets into the shed unless I open the door.
I think your fabric will be fine.