How to seal floor under built in closet?
That's a difficult situation. The best bet might be to caulk everything with silicone from the crawl space, then add some fiberglass batt insulation. 

Otherwise I think I'll look for a heavy, flexible fabric or membrane and lay it in there under the drawers, lapping up the sides just a bit where you can. You could try applying silicone caulk to all joints and seams, maybe from a small squeeze tube, but it'll be messy. You could use rubber gloves and finger the caulk out to press and finish it.