How about the old "Yooper" (Upper Peninsula of Michigan) approach ? It's tried and true for smokers as well as electrode (welding rod) storage. Get a refrigerator that has given up the ghost as far as its original purpose. It is an insulated cabinet. It has adjustable shelves. Hang a lightbulb inside the reefer and leave it on continuously. It will throw just enough heat to knock off the chill and prevent condensation. Plenty of Yoopers use old household reefers and freezers for smokers as well as 'dry rod ovens'. The price should be right for the old reefer, and you will be doing some 'repurposing' as the new buzzword goes.
As for handling lathe chucks themselves: None of us is getting any younger and our backs and fingers are not so easy to fix if damaged. At the powerplant, we had a wide assortment of lathe chucks and faceplates, from maybe 8" on up to somewhere around 24". At some point in the past, the mechanics tapped holes in the rims of the larger chucks. These were used to screw lifting eyes into. We had the luxury of a bridge crane in the plant machine shop, so picking up a heavy chuck and flying it over to the lathes was easily done. The idea of a tapping for lifting eyes in the bodies of the larger and heavier chucks always seemed like a good one to me.
A comnbination of an old reefer and either a hydraulic lift cart or portable hoist would seem like a viable way to handle both the storage/rust issue as well as moving the heavier chucks. Lining the bottom of the old reefer with 3/4" plywood and partially up the sides should insure that the chucks do not break thru the original cabinet liner.
I've been in Yooper shops years ago where there were three (3) old reefers. One had a 3" smoke pipe coming off it, and that was used for smoking fish and wild game. The next had a lead cord coming out of it, and that was the dry/warm storage for welding electrode. The last reefer in the lineup had a lead cord coming out of it, but also had a beer tap sticking out of one side. That was storage for a 1/4 or 1/2 keg of beer, on tap in the shop. Those shops were heated with shop-made wood stoves, sometimes supplemented with a waste crankcase oil drool burner. In cold weather, a speckleware pot of coffee thickened on top of the stove, and a can or two of E 7018 Low Hydrogen electrode might also be found on top of the stove. Seems like ancient history from a past life at this point for me, but the old reefer idea seems like a cheap and easy idea for dry chuck storage. It also provides a clean environment in shops where people are using grinders, welding, chipping slag, needle scaling, and doing less-than-clean shop work along with machine shop work.
