South, East, North?

Well the obvious place is the north pole.

However, you could also go down toward the south pole, and if you start at a position where if you go $10$ miles south, then going east for $10$ miles is essentially walking in a circle and returning to where you started (i.e. the line of latitude at that point is a circle with a 10 mile circumference), then the above would also be satisfied. Its also satisfied if you pick a position such that once you walk $10$ miles south, going west returns you to where you started after going $5$ miles, so doing $2$ such loops would be $10$ miles. And similarly you can choose $\boxed{\mbox{infinitely many}}$ positions chosen such that after going $10$ miles south you "encircle the globe" $n$ times when you walk $10$ miles east, since this will work for any arbitrary value of $n$ .

Note, the solution set consists of one point at the north pole, and then any number of points (a little more than $10$ miles north of the south pole) chosen such that when you go south $10$ miles you wind up at a spot where you will "encircle the earth" an integral number of times in $10$ miles, so walking north again $10$ miles puts you back where you started.