Like other typical stories and novels by William Faulkner, "The Bear" takes place in the southern United States. More specifically, most of the action takes place in the wilderness near Jefferson County, Mississippi. (Part of the story also involves a trip up to Memphis, Tennessee, before the final hunt of the bear.)
How did I know?
Notice how the first paragraph introduces the "camp" where a bear has earned himself a name "in an area almost a hundred miles deep." So we know that the hunting and other action of the story is fairly limited to this area of wilderness. A few paragraphs later, the narrator describes the setting as "unaxed woods," a beautiful way of saying it's an area with trees that haven't been chopped down.
Then, a few paragraphs in, the narrator describes the boy watching people "depart on the road to town, to Jefferson," so we know they're not too far from a town called Jefferson. But lots of US states have a town called Jefferson, so we're not sure yet where the setting is, exactly.
These woods have dear and bears and oaks, so they could be almost anywhere in the south, but there's also a lot of talk about a bayou, so that helps us narrow down where the setting is. And there is also talk of local Chickasaw Native Americans, which we know inhabited Mississippi, Kentucky, Tennessee and Alabama.
Now the evidence is mounting! There are bayous, Chickasaw Native Americans, wooded areas, and a place called Jefferson County in Mississippi. So the story probably takes place near there. In fact, lots of Faulkner's stories are based in and around that area, and he's from Mississippi himself.
For further reading, the American Geographical Society has published a detailed exploration of the setting of this story.