@Michelle Bright,
I would say no.
The expense isn't going to be worth it in the long run due to the damage the floors have sustained. Water damage and pet urine are hard things to overcome when re-finishing a hardwood floor. It looks like some of the boards are warped and will need replaced. By the time you figure the expense in cleaning and prepping the floor, replacing bad spots, sanding, sanding, and sanding again then staining them, then sealing the floors you are going to be looking at a considerable cost that will most likely exceed the cost of Allure flooring.
I recently re-finished a Southern Pine hardwood floor in Owensboro, KY and they were in pretty darn good shape. New owner found them when they went to rip up carpet to put new down. Found the floors underneath and I made 'em shine like new.
Allure will give you the look and texture feel of a real hardwood floor for a fraction of the cost. Most contractors won't want to touch something that is labor intensive like your pictures look like. I could be wrong because I can't touch and smell the wood. But I wouldn't bid lower than other options that's for sure. Mainly just because you don't know what you'll be getting into with the bad spots. May have to tear out sub-floor because it's rotted. Could get messy and expensive.
Take Staton's advice and get some quotes. but I'd would look into Allure Flooring as well. Carpet would be a last resort. Budget considering of course.
Just my two cents. Hope it helps.
Best of Luck to ya!