You can't really "patch" a finish like that.

I have had success scuff-sanding really battered finishes, wiping them with an oil-based stain to refresh the color (pick a color close to the mid-range of the current floor), and then wiping on a layer of floor-grade poluurethane.

Also, look into "screening", which uses a really harsh scrubbing pad on a big heavy machine (giant floor buffer) to knock off the old finish. Then you can stain or just apply a penetrating oil sealant like Waterlox.

It's far easier than wrestling a drum sander, and the possibilities for wrecking your floor are small.

I've posted a link. Scroll down a bit more than halfway, and you will find instructions that will help you to determine what finish you have on your floors.

There's always the chance someone's refinished your floors with polyurethane. But if you still have the original finish, it is probably either shellac or varnish. Those finishes would then have been covered with wax to protect them. The floors would have been cleaned and re-wax once or twice a year.

If you have shellac on your floors, you can refinish just the bare patches. I'd look up directions on how to do this to get all the details, but basically, you clean the patch and surrounding area well, then brush on a coat of shellac. Be aware that shellac comes in different shades, so your new stuff might not be an exact match. Shellac dries quickly.

Then you can, if you want, clean the entire floor and put a coat of wax on it. You used to have to buff the wax after application, but there are waxes out there now that don't require this.

Shellac seals the wood very well, but it isn't very strong. That's why you need the wax on top of it, to protect the shellac from wearing away. I'll bet your bare patches are in high traffic areas?

Several years ago, I rented an apartment in an old tavern built in 1777. The owner had restored things to as close to the time period as he could. My floor was wax over shellac and it was not hard to take care of. And the floor looked beautiful--old hemlock boards that had darkened to a mellow dark honey color.

Here is a link that might be useful: type of finish