consumer-grade modified atmosphere storage?
Air tight buckets and dry ice are used by some preppers to store food in oxygen free environments (as a bonus, it also kills any bugs hiding in the grain/beans).  As long as you make sure that you don't end up with a crust of water ice on the outside you should be able to do the same to keep your items dry and mold free assuming they've been thoroughly cleaned before being put into storage.

What you do is to put a layer of dry ice in the bottom of the bucket, put whatever you want to store on top and then put the lid on almost but not quite tight enough to fully seal the bucket.  Leaving a minimum opening is needed to prevent pressure building up from rupturing the bucket as the ice sublimates.  After the initial partial closing you monitor the temperature of the bottom of the bucket, and seal the kid down when it's no longer extremely cold.  If you're doing a number of containers at once you'll want to make sure your work space is well ventilated.

A 6 gallon storage bucket contains 1.5 cubic feet of volume, and you want enough CO2 to fill the volume about twice over to be confident you've displaced all the existing air inside.  A pound of dry ice will produce 8.3 cubic feet of CO2.  For something like shoes that have a lot of empty space it's probably best to assume the volume if what you're storing in the bucket is negligible for planning purposes, so you'd need about 5.8 ounces per bucket.  At around a dollar/pound it's cheap enough that unless you're storing a lot of stuff at once you'll probably spend more on the gas going to get it than anything else.

Ultimately though, while more practical than Ed Beal's discussion about nitrogen filled warehouses, dehumidifying is still probably the more practical option.

My source for the above was:  http://www.thefoodguys.com/dryice.html