Back focusing and front focusing literally means that the auto focus system incorrectly focuses behind or in front of the intended focus point. An example would be if you focus on the bear, the camera confirms focus and takes the shot, but the focus point is actually behind the bear or in front of it.
In my experience the probability of this is fairly low, but on the newer camera bodies the focus can be adjusted to correct it so it's not a big deal.
If you had the lens all the way out at 250mm and the bears were reasonably close (!), then f/4 would provide a shallow depth of field; meaning that area that objects will be in focus is small.
You might want to try a lower shutter speed like 1/500-1/800 then stop your lens down to f/8 if possible. That's assuming that you can easily see the bears again
Also, it usually helps if you post a picture as an example. Then we might be able to identify and explain the source of the blur.