Oak floor finish

To start, choose an Emory cloth sand paper, the more flexible kind often sold to jewelry hobbyists. Begin with 180 grit and advance to 240 grit.

Sand with the grain (not against it) to about 1 inch over the edge of the bald area, meaning you are sanding a 1-inch transition border around the bald patch.

Sanding will remove any built-up dirt but also could make the sanded area lighter in appearance than the polyurethaned wood surrounding the patch.

If this is the case, test out a few wood stains. You may not be able to get an exact match, as the surrounding floor may have been originally finished with tinted polyurethane, if not stain.

Before you apply stain to the floor, clean the sanded area thoroughly with tack cloth. Then, using a gloved fingertip, apply an inch of finger-wide stain in the bald spot and let dry, following the product's directions. Don't apply much because you may have to sand or feather it out if it's too dark. For the very handy: pop up a door threshold and do the stain test on bare floor boards beneath it (then reinstall the threshold).

Once you know whether the whole patch needs stain, or the floor can be polyurethaned without staining, it's time to choose polyurethane to match the surrounding flooring. There is a little room for error here: if you use flat finish where gloss was needed for a match, or vice versa, you can switch when applying the second coat of polyurethane.

Use a foam brush and apply the polyurethane in a thin coat, brushing inward from the 1-inch overlap on the outside edges into the center of the patch. For my test, I brushed with the grain, but use your judgment since brushing with the grain does not always achieve the desired look. If dust floats into the first coat of polyurethane, sand it out lightly before applying the next coat.

http://www.nola.com/homegarden/index.ssf/2014/08/have_a_worn_spot_on_your_hardw.html

Sand a small area refinis it and compare. You should be able to litely sand and refinish as long as there isn't staining or other damage

after you give up on trying to get it to match throw a rug over it

If you don't want to go to all that trouble, go to the grocery store or home improvement store and buy a bottle of high traffic hardwood polish and restorer. The name of the product may differ; get one for hardwood floors. Mine is for medium colored wood and blends in very well.