What can I do to keep high hydration dough from sticking to my hands?
Let the dough rest in the bowl for a while after mixing. Time will not only improve the flavor (insert autolyse proselytism here) but will also give the flour time to hydrate, making the dough somewhat easier to handle.

Dough will become less sticky as you work it -- witness the way dough kneaded in a mixer or food processor initially sticks to the sides but eventually forms a ball and leaves the bowl clean. So, one option is to work it a bit with a mixer first, or just with a wooden spoon in the bowl.

Anil's suggestion to oil your hands is a good one. If the kind of dough you're making allows it, add some oil or butter to the dough during mixing, too.

Flour helps, of course. Instead of trying to coat your hands in flour, throw a bit on the kneading surface and on the top of the dough.

For very sticky dough, a dough knife can help you scrape the dough off the surface and fold it over onto itself until it becomes easier to handle. This is easiest if you're working on a hard, smooth kneading surface like marble.