.30 cal, weighing 150 grns or + doing 2500 FPS or + will do a good job.
I will shoot Bears I have looked over at a fair distance, if they are running away, as most do. I live on the wide open Tundra, so its a luxury that come with the view. In trees, never more than 100yards if 30, even....
No look over, no shoot, gotta be fat and healthy.
I hunt them in Spring just as they emerge from their holes, and are exceedingly fat, with the fish taste in the meat gone from a long winters nap. Fat at the lean time of year.
I do mostly temple shots, between that little eye and that little ear. The .223 does very well placed in that "spot"
On my use of a .223, the couple times I did use one was because that is all I had. It was also a nail driver, Rem 700 with 6 in the mag and very hand for Fur and Caribou. Worked for Bears Black and Brown, but as a Hunter by profession and filming a sideline, I have ever advantage, and make the most of terrain and surprise. I also know exactly where to place the bullet as shown me by those I pale to in comparison.... Bears dont shoot back but they scoot along pretty fast.... Distance can make that a safer shot, and Im pretty good at doing that.
I guess a semi auto .223 with multipull shots can do the job very well, up close, if accuracy isnt yer thang....
A study the anatomy of the Bear will do you no harm
Were allowed 2 Brown bears every year here, each, but some years I get none, the rest just one. Its all I need, but the skins make for excellent floors in deep winter.
Now for Bear defense, I would pack a 12 gauge.