How can I clean a very dirty concrete basement floor?
Is this a finished concrete floor or raw slab? If slab, I'd suggest low-power pressure wash...maybe even start with just the garden-hose attachment. Add a bit of soap.

The real challenge, however, is that you have no drains. I can't think of an easy way to get around that short of power wash a bit, shop vac it up, and then constantly repeat until the shop vac is full, then empty that into a sink and repeat.