Batteries are chemical by definition. The closest thing to storing energy in nanostructures would be nonchemical supercapacitors, which use nanostructures or nanofabrication to achieve high surface areas. As a side effect of electrical attraction they store some energy mechanically too.

In general storing mechanical energy at the nanoscale is not efficient. As you get smaller mechanical properties become less relevant due to the increase in relative surface area (aka square-cube law). That makes electrical and chemical properties much more powerful.

Formula 1 cars use(d?) flywheels, but that's neither "nano" nor easily applicable to cars. But i think I've also seen it used at a subway in Istanbul.

Muscles are also excellent at storing energy in spring-like fibres that stretch/relax. But I can't think of any manufactured materials at that scale.