It's hard to go wrong with the .30-06 in this country. I hope there is never a time when that cartridge isn't a well regarded and patriotic as apple pie. Though in truth, all of those cartridges would do for what you want.
If you handload, then all the cartridges offer some potential for long range target shooting. If you do not handload, only the .308 and the Creedmore are going to have much for commercial match loads.
I have no experience with the 7-08, but I know that .284 caliber has some of the best long range projectiles available.
Between the .270 and the .30-06, I actually prefer the .270, but only because that was my first hunting rifle and so the cartridge has some sentimental value to me. The .30-06 might be slightly better for elk, moose, and larger game, the .270 might shoot a tad flatter at most hunting ranges and recoil marginally less, but they are close enough you have to nitpick to notice any practical difference. Same with the .308 and the .30-06. Six of one, half a dozen of the other. The .30-06 handles 180 gr and heavier bullets a little better, so depending on how big your bears get, that might be something to consider against the .308, and the .270. The .308 can be had in slightly lighter rifles and will have better factory options and availability for match ammo. Depending on how long of range you want to go, how steep your terrain or out of shape you are, that might be something to consider.
The good news is that if you put the bullet where it needs to be, chances are slim that any deer or bear is going to notice much of a difference.