Based on personal experience, I can say that the .308 AND .303 British AND the .30'06 will all cleanly kill bears and moose and caribou.
This is predicated on using GOOD bullets and placing them properly for maximum effect. (My personal benchmark for bullet performance is the Nosler Partition, but that benchmark is now being threatened by the Barnes TSX.)
An auto-loading .308 loaded with these GOOD bullets is a fearsome instrument against anything. I used a TRW M-14 with 180 Noslers as a camp rifle to deal with barren-ground grizzlies north of the timberline. It worked just fine, and I never worried a bit. Remember too, that a .308 at the muzzle is the full equal of a .300 Magnum at a couple hundred yards.
For charging bears, just get on that trigger and keep shooting while trying for good shot placement. (N0, senator, us civilians have NO NEED for large-capacity magazines... but that M14 with 20 Noslers aboard sure came in handy, and it was a great comfort.)