We bought an old house (1940) and underneath the ugly carpet, there is a hardwood floor. However, the edges of the floor are jagged, as if they had rotted away. Is this a salvageable floor (i.e. refurbished by a professional) or should I just replace it with new?

FYI: House was fumigated for termite before we moved in.

I'd be more worried about how rotted it is under the flooring, at this point!!!!!!!! You may be sitting on a can of worms, that really needs to be opened.

Floorguy is giving good advice, but since we cannot see the floor; we have no way of knowing if the floor is repairable.
Get some bids locally to inspect and bid to re-finish. See what is said by industry professionals that are able to see the floor.