Sanding is the best way; depending on the finish (if its oil based) you can try a light steel wool and a reoiling, but if its been years, then the stain likely penetrates further into the wood.

In short, it can be fixed but any repair would be noticeable compared to the rest of the aged floor. Sorry

You could try a super-saturated solution of oxalic acid. It will take out rust stains and lighten pet stains. It's worth a try , before you sand, but water stains penetrate into the surface so you may never get them out.
Mix as much oxalic acid into almost boiling water as you can and use it hot. It may lighten the area around the stains, so use a brush to a sponge to keep the solution near the stain. It could take multiple applications.

Looks like you need a sand an finish on the whole floor anyway.

Looks like you need a sand an finish on the whole floor anyway....dude.

Buff with #2 steel wool before oiling. You will get a nice old looking floor.
It will take 2 coats to make the shine close to even, buffing after the first coat has hardened.