Nirvana, thanks for the tip! Unfortunately, the problem was sewage coming up into the house (baking soda wouldn't touch it :-) Hours and hours of plumber's work, and ripping out the carpet and padding, cutting away all the wet drywall, and days and days of fans... I think we just about have it taken care of!
Mostly I was completely surprised by the shear quantity of STUFF that was in that closet. Removing it all and "remembering" what was in there was an event in and of itself. It made be realize just how rediculously behind I am....