If you can't build a closet, why not build an aging cabinet?
This idea came to me, you mention about closets being full of clothes. In many other places outside of America, people don't have closets, they have cabinets, wardrobes, etc. They really are a bit more space efficient.
I think that making an aging cabinet of sorts would be sort of a halfway between a refrigerator and a closet, size-wise.
You could just build it as a regular wooden carcass-type cabinet, and put some foil lined polyurethane insulation on the inside, then proceed to cool/humidify like you would a closet type cave.
Nuts, now I have the urge to build such a contraption.