Solid Hardwood, Engineered and Laminate Flooring - Stains

My wife and I just finished ripping up the dog-stained carpet in our new house to find the stains are set deeply into the wood floors. One in particular seems to have been the dog's favorite spot. It is very dark and STILL smells as if it were made an hour ago. My question is, is there a way to get rid of the smell in the wood without sanding it? We plan to replace the floors when we can afford it but for now we're going to paint them. But I don't want to smell that and I'm worried that the paint might not be enough to cover the smell. We've left the windows open overnight to air out the house, is there anything else we can do to speed up that process? I don't want a temporary solution like an air freshener. Thanks in advance to anyone who can help me with this problem.

You could try bleach, but instead of chemicals, I would just replace the wood. Sometimes its bad enough that its penetrated the subfloor as well.

If you plan on painting floors, cover stains with a couple coats of primer/sealer like Zinnser to seal the stains and odor. Pet urine that has penetrated to the subfloor will remain.

We were planning to cover the paint with some sealeranyway, so it's good to hear that this could help.

As far as replacing the boards, we're moving in three days and to put it bluntly, we're flat broke for the next few weeks. We'll barely be able to afford the floor paint.

I'm not too worried about the subfloor, though. The floor paint will be a temporary solution for about a year or so and then we'll probably be ripping it up down to the joists anyway and laying new plywood before we put whatever we decide to put down.

I'll definitely try the bleach, though! The smell seemed to mostly go away when we went there to work at 4PM today, so I think airing it out helped. Bleach will probably do the trick.