The tuberous roots of spider plants do store water, but not as much as a Cactus stores water in its stems, for example. Spider plant tubers help the plant survive during brief periods of drought, but the price is usually a lot of discolored leaves and dieback of foliage.
I doubt if anyone has tried growing a spider plant in moss. If you decide to try it, let us know, preferably with some photos, as to how well it worked. We can all learn.