Because of peculiarities in the cartridge and the rifle itself, it can't be loaded "hot" and the muzzle energy is far below the .30'06 Springfield and even somewhat below the .308 Winchester. I have heard it described as about equivalent to the .30-40 Krag - but I have never owned or fired one of those.
While the .303 has been used on dangerous game in Africa, I'm inclined to think that the persons involved just got lucky, and for every lucky fellow there are probably a hundred unsung nebbishes who wound up being grease under the hooves of a Cape or between the toes of an elephant.
Absolute minimum for any bear larger than ordinary black bear in the lower 48 (for which .30'06 or .308 are perfectly adequate in the hands of a decent shot) would seem to me to be at *least* a Remington 7mm magnum, or one of the old big, heavy cartridges like a .415 Rigby. Just about all the pro guides in Africa now carry .458 Win Mag - just in case.
My dad bought a .348 Win (one of your old big heavy brush-bushing cartridges) for my mom when they were planning to hunt Kodiak bear in Alaska. He was about to buy a .375 H&H for himself when they turned up pregnant with yours truly and the hunt had to be scrubbed.
He used to look at me occasionally, shake his head, and ask sorrowfully, "Why couldn't you have waited til I got the H&H?"
But he was a remarkably good shot.