Seven layers of linoleum on the kitchen floor held down with some kind of mastic and nails. Just as well there was really no kitchen so we could tackle the problem head on. We got out putty knives and started to pull, scrape and fill barrels with this junk. There were tiny nails everywhere. The final part and the most difficult was getting the black, hard as a rock stuff off the wood itself.
Kept scraping, then used a sander to finish off the wood. Very few stains in the pine but still a million tiny holes. Just used a wood bleach on the few stains and they lightened up enough so we could finish off the job. All these years later, the floor still looks good. I took the suggestion of DUer and used Minwax Floor Refinisher as a refresher a few years ago and it has held up well.
As to a new floor, are you sure you have clearance under the doors to accommodate another layer of wood?
I'd get a hefty hand held sander with lots of sandpaper of various grades and go at it. A good vibrating sander and supplies would cost far less than a new floor or a professional to do the job. You have to be patient and have a good back. Down to the bare wood, stain and poly.
Although the kitchen floor was the hardest to do for us, the other rooms were more frustrating. It seems one or many of the prior owners just painted the floors around the room size rugs. BLACK! That was another process. I think I hated that with a passion. Try to get a refinished area to match up with the aged floor underneath. Worth it in the end.
This old Yankee never gives up.
BYW: Have you checked places like Craigslist or your local paper to see if there is someone within a reasonable distance who could do the initial --or all-- the work? I bet there is someone who has the skill and needs the work.