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.
Sub Floor
Sub floor material can depend on what type of floor is going to be installed on top of it, and the joist spacing.
Typically if the joists are 16" on center (OC) or less, you can use 5/8" tongue and groove plywood glued and screwed to the joists. larger joist spacing calls for 3/4" tongue and groove plywood, again glued and screwed to the joists.
If the flooring material of choice is a heavy tile, you'll want to use 3/4" tongue and groove plywood even with 16" OC joists.
Underlayment
Depending on what type of floor covering you've chosen, an underlayment may also be necessary.
If you are planning on installing carpet, no extra materials are required.
If you plan to install hardwood flooring, you can use felt or rosin paper as an underlayment (though not all installers use it).
If you are installing laminate flooring, you should use a foam underlayment.
If you are going to install tile, you'll want to use a concrete board or similar material on top of the sub floor.
If you want to install vinyl or fiberglass sheeting, you should use 1/4" Lauan as an underlayment.
Best Answer
I would find it hard to believe that someone would sand a wood floor but then neglect to put a protective finish on it (unless this is a very rustic building like a barn?). It's possible there's varnish there but it's not a glossy type. An unfinished wood floor seems like a negligent decision by the landlord and a bad investment on their part. I would check with him/her. I don't know where you live or what the landlord/tenant laws are, but in general at the end of your lease you are not required to leave the property in pristine condition: normal "wear and tear" (for which you are not responsible) would be considerably higher for an unfinished floor.
In any event, the best thing you can do for any wood floor (regardless of finish) is to keep it clean. If you let sand and dirt accumulate they will act like sandpaper and wear the surface away much more quickly.