Based on information compiled over the past several years by ammoland, it seems pretty conclusive that contrary to popular belief, Bears are not in fact four-legged Sherman Tanks that can only be dropped by a .50BMG APIT round, but rather creatures of flesh and blood which can and have been killed by literally every cartridge on the market including .22LR.
So yeah, unfortunately campfire stories about the Bald-headed Bear of Claire County just don't hold up in the information age. You don't need a .500 Magnum Revolver or .338 Lapua Magnum Rifle to drop a bear, you can do it with whatever you might have handy.
Granted, "can" and "will assuredly" are two completely different things and a more powerful cartridge is more likely to succeed, so definitely carry the most powerful thing that you can use proficiently, but other than that, you're probably fine.
.357 Magnum is obviously a powerful and very capable cartridge which if loaded with solid, non-expandind bullets most certainly has the necessary energy as well as sectional density to penetrate a Bear's skull and shut it down.
If all else fails, you could always just stay out of Bear Country, then your likelihood of ever having to shoot a Bear will decrease dramatically.