Thanks for playing spykemajeska, but Moonray just won with the following answer: One way this is possible is if you are on a line of latitude so close to the South Pole that the entire circle of latitude is exactly one mile around.
To begin, we travel one mile south to point on the circle. Then, we travel east along the circle, and by its construction, we end up exactly where we began. Now we travel one mile north, and we reach the starting point of the journey, exactly as we wanted.

My response to him: You win Moonray. In fact there are an infinite number of points on the "ring" where you can start from. In fact there are an infinite number of similar rings. The second one a little South of what you described where after you go South 1 mile, when you go east you go "around" the earth in an easterly direction two times as it is 0.5 miles around. Third sub ring is 1/3 mile around, etc. You may want to have a heavy fur coat and some snow shoes however! You got my upvote. Congratulations. I have another riddle coming up in a bit...keep tuned.