I live and hunt in Alaska where there is a always a chance of shooting a big bear whether you want to or not. Since I was like 10 I've been packing a .308, before that it was a 30-30, and before that I would go hunting with dad and pack a .22 around. He carries a .338 win mag.
I just switched to a .30'06. I feel perfectly safe wherever I go. Bullet placement means a lot more than bullet size, so shoot a lot and know where your bullet will hit. If you get into a situation where you need to stop something BIG and you need to stop it NOW, you want the biggest cannon you can carry. I, however, want to shoot my hunting rifle a lot without being beat to death by it, and as a .308 or 30'06 can get the job done and have proven it time and again, they'll work for me.
So get your .308, if you're still worried, get a 30'06. It will perform better with bigger bullets. Just try to drop the critters before they get their adrenaline up. If you take your shots while an animal is bedded down or feeding, you can take almost any big game animal with almost any cartridge.