From the perspective of a mechanical watch, there is a component known as the mainspring. It is a spring that stores energy from either manual winding or automatic winding.
With manual winding, turning the watch's crown to wind the mainspring is an intermittent, rotary motion. Alternatively, with automatic winding, a rotary weight that moves intermittently (e.g., when the wearer's arm is moved) is used to wind the mainspring.
In either winding scenario, manual or automatic, the result is that intermittent, rotary motion causes energy to be stored in the mainspring, which is then subsequently available to provide for continuous rotation of the watch's mechanics.
I caught myself thinking along the lines of "of course it's not possible, that would be preposterous!", but as you say, a watch spring mechanism does exactly this.
The key concept here is "stores energy".