Answer to old chestnut: 50 miles south, west, north
The answer depends on the curvature of the earth and polar coordinate systems. Normally you can ignore the curvature of the earth on such a short trip, or at least, it won't make such a dramatic difference as in this problem.
The traditional answer is that the man's house was on the north pole. All directions leaving the north pole are south, so he left his house in any direction, traveled west a while, and went back north to his house (any journey north from anywhere will eventually lead to the pole). Because of this, the bear the hunter shot would have been a polar bear, and thus, white.
There is another set of possible answers that puts the man near the south pole. He needs to live 50 miles north of a point where the trip west will represent one or more full circles around the south pole, bringing him back to the same spot before he begins his trek back north.