Contrary to the other responses, I have had success by refinishing/resurfacing floors with such stains. However, a great deal depends on the type of hardwood you have, and how long the area was allowed to stay wet. Also, you were talking about an area rug. Many of these contain dyes that run. Your stain may be more from these dyes than from the urine, which can be much more difficult to deal with. I have also had success with some chemical combinations, but their results may vary depending on material and finish. It would be hard to determine without seeing the floor.
Although sanding has worked in my experiences, it can be a rather drastic measure. As a DIY project, it can be messy, disruptive, and somewhat time consuming because it involves resurfacing the entire room, or more if the hardwood continues beyond. If you hire someone, itís messy, disruptive, and somewhat expensive. I would consider refinishing as a last resort, unless the general condition of the floor warrants it. And after youíre done, there is no guarantee the stain will be gone.
So before you do anything, get a couple of opinions from flooring specialists in your area. They may recommend a cleaner you may not have tried, or maybe staining the floor a darker color will hide it. Maybe replacing the wood is not that big of a deal because of the stainís location. At any rate, they should be willing to come out and evaluate the stain and floor in general, then give you some options to think about.