Bears take four hits to kill. Unlike other enemies, they are not instantly killed by spear traps; a spear trap only counts as a single hit. However, arrows also count as a single hit.
The safest way to kill a bear is to make a gas bomb and chuck it at the bear, then shoot it four times with arrows. You can also set up a couple spear traps, shoot the bear with an arrow, then run through your spear traps and shoot at the bear while it struggles through them.
If there are snakes around, luring the snakes to the bear using the torch, then baiting the bear to go through the snakes can also deal it damage.