Clean (food) oil stain from wooden floor
Fisrt, DON'T REPLACE THE FLOOR, that would lead to a lot of headaches and is probably un-nessassary.  That floor shouldn't absorb oil that quickly if it had been sealed properly.  My guess is that the floor either wasn't sealed correctly or it's been so long since it has been that it's all been worn off.  

Try getting a wood oil soap (like Murphy's or Old English) these soaps have oils in them that are meant to be absorbed into wood. This won't take out the oil, but it might help to make it blend in.  I don't think that stain is comming out, even if you sanded the floor, the stain would have absorbed into the wood too deeply.

It's important to remember that wood is pourous and the natural oils that are in it are evaporated over time and have to be replaced, or else the wood becomes discolored and brittle, which is probably the case here.  I wash all my wood floors twice a year with an oil soap to help keep them hydrated.