Is this a regular floor drain like a shower drain, or is it a drain in the wall where the water supply comes out? Try obscene amounts of drano/liquid plumr. Does it go to a sewer line? Can you get to the drain below you in a cellar, basement or crawl space? If so, does it have a P drain in it? That's usually where it clogs. PVC or cast iron?
If you set up a laundry sink and it's a direct line to the drain, then the basin could possibly hold the excess water until it has time to drain.
Good luck. Buying and installing the laundry sink might be more expensive than a plumber.
RTR,
Tim