Hello,
I handle vintage & antique garments for a living, and I've worked with museums (including the Smithsonian) for the last 11 years. It is important that the items be clean & dry, but it's also important with natural fibers to allow the items to "breathe" (expand & contract). If you are storing only synthetic items, it's not important to allow them to breathe, so in this case you can seal them airtight.
To avoid yellowing, I wrap items in unbleached 100% cotton muslin that's been washed at least once. If the items are well-wrapped, you can store them in cardboard, wood, etc without worrying that the acid of the container will affect the item itself. I also recommend that the item be checked every year for any evidence of pests. With durable fabrics, some people like to wash the item yearly also. This will eliminate any yellowing that could otherwise build up over years sitting unwashed.
Hope this helps!