The simpler question would have been what gun, but instead it says rifle. A 16ga pump is a nice compromise as an all around gun, 12 and 20 work too but 12 is too heavy at times and I would want more oomph than a 20 for brown bear.
So back to this rifle conundrum. The thought strikes me that everybody went bolt action. A .50 smoke pole would work too as would a 45-70 lever gun. As always with this question, your giving up a lot to try to gain it all. It's like trying to pick just one hammer. Sure a 12 oz claw will drive a nail but you may need to break rocks, so it's going to be hell driving nails with a 12 pound 4 ft handled sledge. Pretty much any center fire will work, as would most rim fires with shot placement and luck being key. The answer here will end up as 30-06 but others will do just as well. 308 is a short 30-06. 270 is the same as a 30-06 just smaller lighter bullet moving faster and flatter. 7mm is not very efficient but will do it...yep pretty much anything from 22-250 to 50bmg can be argued intelligently but it all comes down to your preference. Honestly, if I had to pick one rifle it would be a .270 or a 6.8. Most will pick a dirty thirty. In brush country there will be folks picking 30-30 35rem 44mag 45-70 etc.
Pick your action based on terrain. Pick your caliber based upon versatility and variety of critters in that terrain. Then go shoot them and eat them.
