na-adjective
Describes a state of having nothing to do and feeling restless or out of place. Often used with そう or に to describe someone's appearance or manner.
See also: 所在無い
彼は所在なげに窓の外を眺めていた。
He was idly gazing out the window.
会議が長引いて、みんな所在なげそうにしていた。
The meeting dragged on, and everyone looked bored and restless.
所在無い is the dictionary form of the adjective meaning 'having nothing to do; bored'. 所在なげ is the -げ form, meaning 'looking like one has nothing to do; seeming bored'.
Derived from 所在 (whereabouts, something to do) + 無い (not exist) → 所在無い (having nothing to do). The -げ form adds the nuance of 'appearing to be'.