I haven't seen Pirates of the Caribbean in at least a decade, so forgive my ignorance as I don't quite remember the scene, but there are a few cases I can think of when someone would "get their boats up on shore."

First off, most maintenance in the period I study would be performed by docking a ship, usually in a drydock of some sort. King Henry VII built the first permanent drydock in England at Portsmouth, in 1495 (it is still in use). If there wasn't a drydock available, ships could be pulled into shallow water or next to a dock and careened (that is, tipped partially over, very carefully) so that repairs could be made to their bottoms.

What you're describing, though, sounds like a ship has been pulled up entirely on shore. It might be practicable to do that with a small sailing vessel by using something like logs as rollers -- we know that triremes were stored on land in ancient times, for example -- but the actual drawing of the ship out of water would have required quite a bit of muscle power. All but the smallest sailing vessels during this time period were fitted with capstans to provide some mechanical advantage to winding lines.

EDIT: I just remembered there was a similar question asked about the movie version of Les Miserables in which a group of prisoners are seen dragging a ship into a drydock; I answered it here.