The hunter's door is in one of two locations. One is a foot or so from the North Pole, facing north, such that his position in front of the door is precisely upon the North Pole. Since that's a ridiculous place to build a house and since bears do not roam within fifty miles of the pole, the bear is either imaginary or imported, and there is no telling what color it is.
There is another place (actually a whole set) on earth from which one can go fifty miles south, fifty miles west, and fifty miles north and end up where one started. Consider the parallel of latitude close enough to the South Pole that its length is 50/n miles, for some integer n.