Specific kind of plastic that won't make fabrics smell? I store various fabrics (clothing, cloths, linens, etc.) in RubberMaid RoughNeck plastic bins.  It seems to work fine for a while, but given enough years, the contents take on a strong chemical smell.  These are not stored in a musty or damp area -- they are stored in my living room.  I assume, therefore, that the smell is from the plastic.  Before using the contents again, I need to wash them.  This means competing for machines in the laundry room of a highrise.  This entirely defeats the point of keeping the seldomly used items, which was to have them on stand-by for immediate use upon demand (as opposed to going out to re-buy them).

Are there specific types of plastic that I should look for in a bin to avoid this kind of smell from the plastic?
It is likely a plasticizer in polyethylene bins, rather than polyethylene itself, that has an odor. Storing the garments in clear polyethylene plastic bags (perhaps inside the bins), which have different plasticizers, should reduce the odor -- and some bags are designed to compress clothing with use of a vacuum cleaner, saving space. 

Another option is to get an inexpensive wooden (cedar lined?) chest, or a cloth storage bag. Advantage: breathable. Disadvantage: does not protect from moisture.

BTW, some of the plasticizers include polybutylene, dioctyl  phthalate, glyceryl tribenzoate, polyethylene glycol, sunflower oil etc. Handbook of Plasticizers Ch. 11, G. Wypych. Perhaps rancid sunflower oil is the odor?