RE: Is .308 enough gun
The 308 is a dandy round and one of my favorites for deer and is a good round for elk and such if used within it's limitations. On large animls such as moose or elk I'd keep shots 200 yards and under with a well constructed 180 grain bullet.
Can you kill a Brownie with a 308 ? Sure ,but a brown bear is a tuff customer and depending on time of year the bullet has to punch thru fat ,muscle and rib cage bone at the least. It has to deliver alot of energy so a shot IMHO with a 308 would have to be in the 125 yard range max.
For the great bear I believe it is a marginal round for all but the experienced marksman/hunter. I believe a 30o mag is a better choice with the 338 and 375 H & H my top vote getters for bears.