I would take a shot at a mountain lion with a .223, with a couple of caveats. First off, I'd use one of the heavier bullets intended for deer, for instance. I forget the brand, but my slow-twist Ruger is accurate with a round-nosed 70-grain bullet.
The other point is to shoot at a lion that's not moving, or is just walking along slowly. "Way out yonder" would be a no-no, as would be a shot at a running lion.
(A just-spooked lion can go a hundred yards in maybe four or five seconds. Odds are, a running shot is a waste of time.)