We did look into having the floors sanded and refinished but this is what we found:
Older homes tend to do better because they used a different kind of finish. This is why my friend with the 100 year old home has beautiful floors but our house is only about 50 years old. Also, our floors are really light as is all the woodwork so staining dark would look incongruous. Finally, these are really old repeated stains (there is a black trail from one end of the room to the other) and it’s likely that they went through the floors completely meaning that we would actually have to replace about 1/3 to 1/2 of the floor making it out of our price range. We did have one friend who recommended that we paint the floors but I don’t particularly like the look.

More than you wanted to know, eh?

The stains had not gone through carpet - the floors were just wood - so that might make a difference, but:

I hired a floor refinisher. It was d______d expensive, but worth it. He came in and sanded down all the floors (this was before I moved in), restained them with a darkish color, and sealed them.

The pet stains went bye-bye. I think the sanding was the trick - he sanded enough to get them out but not enough to really hurt the floors (this is an old, old house with good red oak floors).

Or, you could do like one of my colleagues and buy lots of throw rugs to cover them - if you don’t mind having throw rugs in odd places.