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: