Specific kind of plastic that won't make fabrics smell?
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?