Can I hire a non-Jew to come to my house on Shabbos to help me stretch? I enjoy stretching, and it helps me relax. Payment is made during the week.
Yes, if no violation of the Shabbos is involved in the work he does for you, there is no problem of him coming to help you on Shabbos.
This is true if, as you write, the stretching is for relaxation purposes (and not if it is for medical purposes).
For the question of stretching on Shabbos, see here concerning massage; the question of stretching is more simple in leniency.
Concerning payment of the non-Jew, there is no problem of paying a non-Jew after Shabbos for his work (where the work involves no melachah).