I do not consider a 357 adequate to hunt bears. Not saying it won't kill one. Bears are an animal that most of us won't shoot often. Bear bones are heavier that deer bones. Bear hide is thicker than deer hide. Bears tend to have a bunch of fat under the skin where I have hunted them. Bear hair mats with fat and blood and stops up wounds. Even well hit bears often dont bleed well. Bears don't have sharp hooves to make easily found tracks. Wounded bears can be unbelievably hard to track if you don't have snow.
A proper 357 bullet will probably penetrate deep enough to cause the bear to die. The problem in the places I hunt are finding the bear. If you are hound hunting and the bears don't run big, probably will work, but the dogs might get scratched up. If you live where bears get big and fat, a bigger gun should be used, in my opinion. You owe it to te bear and yourself to carry a bit more gun, again, in my opinion.
Bfly
