How do I clean a really dirty really old hardwood floor?

Looks like it is ready for a professional floor sander for the whole thing, , , However you might try a disk, orbital, or belt sander VERY gently around the edges. . . Always try a hidden area before tackling the whole place. . . . .

Here is a really good article on cleaning and care for hardwood floors. If you are not ready to Sand them and start fresh, start cleaning them and see what you end up with. Hopefully some really good cleaning will bring them back to life. Good luck!! https://www.thisoldhouse.com/ideas/how-to-clean-wood-floors

They used to stain the boards around a rug years ago, maybe it is Stain. Whatever, you will need to sand ut down with a machine to get it right.

Murphy's oil soap does wonders. Or, if it is really tough, try PB Blaster. Get it at a big box hardware store, like Lowes. Spray, let sit 10 minutes or so, then scrape with kitchen scraper, or old credit card. Spot test first. Then give the whole floor a good mopping with very hot water, mop wrung nearly dry.

have you tried Murphy's oil soap?

https://www.doityourself.com/stry/floor-refinishing-without-sanding

One thing that is around the baseboards is probably mop water and floor wax built up over many years. Strip the whole floor if you can. If that’s too big a job, buy some Pledge Floor Care that dissolves wax. It may take some scrubbing with hot sudsy water and a scrub brush. You may not be able to remove all the built up gunk the first time. When you are finished, you could sand the whole floor down and stain and seal it, or just clean the whole floor with bleach water to brighten it. Then apply Pledge Floor Care to make the floor shine and be easier to clean the next time. PS: you could replace the baseboards without redoing the floors with stain and sealer until you are ready. 😇

I lived in an apartment like that. The floor was so old it looked black in spots. If you can stand it. I used a mop with ammonia in water and a little dish soap. Be careful though. Straight ammonia will strip the finish right off the wood if you leave it on too long.

Why don’t you put vinegar in a spray bottle, spray the hell out of it, come back with a toothbrush later & see what happens

http://www.noordinaryhomestead.com/deep-cleaning-old-wooden-floors/

https://www.woodfloordoctor.com/restoring-beauty-to-old-dirty-hardwood-floors/

I have used PB Blaster to remove industrial tile adhesive from a wood floor. Think WD-40, times 10. I love it. I keep a can on hand to un-stick most things, remove gum, tar, glue. Great penetrating oil.

Take pix and document all spots and stuff so you can get deposit back when you leave !!!!!!

Stop worrying about it - leave it - COVER IT UP! If generations of others have tried and failed, you are unlikely to make a difference in this case. I am most certainly not a quitter, but if the owner isn't bothered , I would say, why should I. Paint over the floor or cover it in anyway you wish, but don't waste anymore time on it. Best of luck...........

Hopefully the Murphy's Oil Soap worked, it is probably the best product out there for problems concerning wood cleaning.