First, a quick note on technique - start with a coarse grade sandpaper on the floor sander and work down to the the finer grade.
The coarse grade removes the old finish and levels out the imperfections, then the fine grade makes it nice and smooth. If you go the other way around, you'll definitely curse every time you see the grooves in the sunlight. Fixing this means stripping off the nice new finish, not fun.
In passing, an obvious-in-hindsight but sometimes-missed point is to go with the grain, i.e. sand along the boards, not across them. In any case your sander will complain bitterly if you don't.
Also, give the boards an incredibly good wash before you start with e.g. Sugar Soap - you want to try and strip off everything gungy to stop the sandpaper clogging too often. Leave for at least 24 hours to dry nicely.
Be sure to hire an "edge sander" too, this will help you get to the edges that a drum sander won't get to. If you can, remove the skirting boards (baseboards) as then you can get right underneath; only do this if you know you're redecorating the walls too and don't mind patching up the woodwork. You'll inevitably end up doing the fiddly bits around the door frame and right in the corners yourself by hand with a scraper or sandpaper, which takes longer than you could possibly imagine.
While doing the sanding, remove everything from the room (obviously) and get heavyweight dust sheets to tape around the door. Wear airtight goggles as this is an insanely dusty process. You'll also need ear defenders as this is an insanely loud process too.
Regarding stain, it's a matter of personal preference. I've done both and prefer the more uniform result that stain gives.
Regarding finish, I've used both oil-based and water-based finishes. Both are OK, but the oil-based one looked richer (can't say why, can't say how) -- and took about three days to dry. Somewhat inconvenient for a hall, and our puppy left nice pawprints.
Follow the instructions on the tin; in general thinning probably isn't worth it, as you'll find the second and subsequent coats will go further anyway. I applied three coats of the oil-based finish, and five of the water-based finish, but this depends on the state of your boards. It's handy to rent good bright work lights so you can see where you've been, especially for the second and subsequent coats.
What is important is that you get flooring-grade varnish, regular varnish won't stand up to the first party, and yacht varnish will cost so much you won't have the first party...
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.
Best Answer
If the surface is edge grain and the previous finish was only wax, wiping down with mineral spirits and then sanding using progressively finer grits up to 220 or so should be pretty good preparation to refinish. Skip the soap and water treatment as it won't dissolve any wax and only complicates finishing. Before applying any finish the wood should be smooth and dry. Any staining or blotching you can see is not going to be hidden too much by a clear finish - it's sometimes possible to bleach out stains but this is tricky to do well.
If you are still getting something soaking back up after sanding down to the raw wood, it could be a prior mineral oil treatment. Mineral oil never hardens and is often used on wooden butcher blocks and countertops. Simply reapplying more would be one refinishing option. It's easy (you just apply until it is no longer absorbed) but you have to reapply it frequently. Inexpensive pure mineral oil can be found in a pharmacy (it's sold as a laxative).
Another option would be a pure tung oil finish. It would be slightly more expensive but a little more water resistant. You can get it from a woodworking or finishing supplier. Most "tung oil finishes" sold by hardware stores are not actually tung oil so check the labels carefully if you go this route. It would be applied and reapplied similarly to mineral oil.
Some "Danish oil" finishes are a blend of varnish, oil, and thinner, and others contain only thinned varnish. Generally one applies "Danish oil" in light coats such that a film is not really built up on the surface of the wood. Often wax is then applied over it. This feels nice on furniture but it is going to be less durable that a slightly thicker application of regular varnish applied with a brush or thinned and then wiped on in several coats.
The mineral oil or tung oil would be my choices for a counter that you are actually going to cut on. On the other hand if you don't plan to cut with knives directly on the counter, and if you are able to prepare the counter down to a clean, dry, smooth surface, a wiping varnish is probably the most durable and best looking option.