Long barrel for sure, but a bear pistol - not likely at all. Even if .50 cal. I would not call it a bear pistol.
IMHO, 'bear pistol' is just a name given to these long barreled pistols. There does not seem to be any early records showing these type pistols were ever used to hunt bear. I think it is a more modern moniker. I certainly would NEVER shoot a bear with a .38 cal. anything.
CH
(Spock eyebrow raise)
It was my understanding that family tradition at least linked the Matt Gillespie and the Montreville Plott pistols with bear hunting. The Plott pistol has a comparatively short barrel, but the Gillespie piece fits the usual criteria for a "bear pistol" pretty well, and is where the moniker comes from, I think.
As for caliber considerations, I don't think what someone might do today while hunting under modern game laws and fair chase conventions have much bearing on what earlier generations might use to dispatch black bears treed by packs of hounds or caught in traps (whether steel legholds or wooden pens). I don't want to risk the wrath of the moderators be specifically bring up modern cartridges, but I think it is pretty evident that up until a couple generations ago there were a lot of folks that were comfortable hunting big game, even bear, using loads that would be considered woefully inadequate today. As for why, I suspect that better levels of marksmanship and stalking skills, different hunting techniques, less concern about potential suffering on the part of the prey, and less (or no) influence by commercial interests competing for the ammunition market by hawking more and more powerful products were all factors.
In other words, I bet that if you are in a position to walk right up to the bear and put the muzzle to its ear, a .38 pistol works just fine...