How do I remove dark spots from old hardwood floors ?

Hi Marilee, I would start by thoroughly cleaning and then sanding the entire floor down. At that point you'll be able to assess if they're deeper than a sander can reach. You could consider staining your floors dark after that to disguise them. After that here's an article that might help pull some of those stains out: https://todayshomeowner.com/video/tips-on-removing-stains-from-wood-floors/

After a deep clean, if you don't want to sand you can use Rejuvenate. It's a great product.

If they are real hardwood they are worth the effort to restore. Rent an orbital sander and redo the whole floor. Remove all furniture etc and block off the entrances to the room with plastic or your house will be covered in dust. It takes less time than most people think to redo a floor. I had 100+ year old planking and redid the entire house. Best advice, keep the sander in motion, never let it sit in one area, just keep it going. Clean the dust thoroughly. I did two coats of stain and two of poly with a rag and a wide brush for the poly. When you use a rag to stain you control the spread and the color by being able to apply just the right amount and can spread it before it soaks in too fast. Don't use too much on the rag. If you can't do the entire floor you can spot sand areas and I will be you are correct about the dark spots being pet stains. I have done around an area rug with spot sanding and redoing by rolling up the edges of the carpet and just giving the floor a new coat of stain and poly. If it is real hardwood, take the time to redo the whole thing.

Before I redid my hardwood floors I poured hydrogen peroxide on my pet stains. It did pull a few out but mostly it just lightened the stains (some considerably). Need to let it sit for 24 hours. Then wipe up. Good luck. Time consuming but worth it.