So your house was probably built with subfloor and hardwood or pine on top of it. A reasonably common thing to do is to strip the hardwood and use the subfloor as your new surface. It's a cost effective way to get rid of really old flooring that may not be salvageable. At the end of the day, a century is a long time for wood flooring to stand up to daily use.

You can certainly refinish subflooring. My recommendation is no stain, just a clear coat, but I personally love lighter tone flooring. It is a softwood, so keep in mind that it will not hold up against children and pets. There's also probably bigger gaps between the boards than you would want.

At the end of the day, the best thing to do is to lay down new hardwood after securing the old boards snuggly onto the joists. But that's up to you.