An other possibility would be a 2nd referendum but then you sort of have the same democratic problem (you can't keep asking the same question until you get the answer you want) and on top of that you'd probably need three options if you held a referendum today: no brexit, soft brexit with the current proposal or hard brexit. I don't think any of these three options could get an absolute majority so I don't really see how you'd deal with the outcome of such a vote, whatever it may be.
I guess the lesson to all the democracies in the world is that you shouldn't ask a question if you're not able to deal with all the potential outcomes. Cameron held the referendum because he fully expected "remain" to win and he quit as soon as he lost. Since then it's a shitshow because nobody can agree on what "brexit" means exactly.
At this point I think we should all brace ourselves for a hard brexit scenario, a big jump into the unknown for everyone involved.