It's only certain plastics that are a problem. Plastic bags are generally not a good idea; they outgas chemicals that can eventually damage fabric. It may be only some types of flexible plastic; I'm not sure which.
My understanding is that solid plastic containers are fine. The only problem with them is the possibility of sealing in moisture along with the fabric (which could eventually produce mold). A few holes in the lid or body means there will be some air circulation and a means for moisture to escape. I do wonder, though, if they would give mice a means of getting in and out of the box.... :shock: