There's no way to stand still per se, but we can tell the difference between someone who accelerates and someone who doesn't - that's the resolution of the "twin paradox". So if one person travels around in a circle and their twin stays still (by magically floating above the earth without following its rotation, or by staying suspended at a particular point in the earth's orbit for a year while the earth goes around), then they have accelerated less than their twin and should therefore have aged slightly more.

What I don't understand is: does this still apply under General Relativity, or does proximity to the massive earth redefine acceleration?