I have a couple of old axe sharpening files that can be used on very dull hatchets. I don't need them often, unless a friend hauls out a beaten-up or neglected tool like you describe. I expect that a regular sharpening stone would be very slow and tedious under the circumstances you describe. In your situation I think I'd start with a file, then move to a stone (or other finer sharpener).
When you are using a file on a hatchet, I would keep it relatively flat to the side of the head. You can adjust the final edge angle when you get it cleaned up enough to transition to a finer sharpening stone. [Of course, if you know someone with a decent belt sander, problem solved!]
Best of luck with restoring it,
DancesWithKnives