The North Pole is one such place that you can walk one mile South, one mile East, one mile North and end up where you started.
Somewhere near the South pole, there is also a point where once you walk one mile South, you are standing at the point on the globe where the Earth's circumference is one mile. That way once you walk one mile East, you arrive back to where you started walking East. Of course, one mile North from there puts you back at your starting point.
In fact, there are an infinite number of points where this is true. Take the preceding case, but instead of using the place where the circumference of the earth is one mile, use the place where the circumference of the earth is 1/2 mile and start one mile north from there. When you walk one mile East, you simply walk around the globe twice and end up in the same spot. This will work for a circumference of 1/3 mile, 1/4 mile, 1/5 mile, etc to infinity.