IMO!................NOPE! Not a 4" 357 mag.
Black bears,,,,yes!......."BIG" BEARS??,,,,Absolutely NOT!!!
A 44 mag imo, would be a bare minimum for the big bears.
Much better back-ups for the big bears are the,,,454 Casull,,,460 S&W,,,500 S&W,,,and if you`re a single action fan, a 45/70 is available from "Magnum Research" along with the above choices from Magnum Research as well. They are BIG!!!
A handier 4" barreled double actioned 500 S&W would be my top pick (I gots one), followed by the S&W one revolver two cartridge combination of the 460 S&W and 454 Casull in the same barrel length.
A 357 for big bears????..........NO! NO! & NO some more!!!!