Bears are a non-elemental danger in sandbox mode. They wander around in Pleasant Valley, seeming to stay near areas with caves. They are also seen in Coastal Highway, wandering along the logging routes, the highway, and even on the lake. Unlike wolf encounters, bear encounters do not display as a combat situation (with a fight meter on the bottom, and defensive weapon shown in the bottom left corner), making it impossible to fight off a bear. Also unlike wolves, it is undocumented whether bears actively hunt other animals, or even feed off of carcasses.
Bears can easily be outrun if you start running before they get too close, but once they are on you, it is extremely hard to survive. Play dead (stop clicking) and you might survive, barely.
It takes several bullets to kill one, and the most efficient way is to shoot it 2-3 (max. 5) times and run while it bleeds out.
As a warning, Bears seem to be exempt from the rule, as they will still attack you in Pilgrim difficulty if shot at, so it's best to keep your distance if you aren't interested in ending your peaceful walk through The Long Dark.
Bears can be skinned for: