하염하염없이 is translated by naver dic as blankly, vacantly:
그는 하염없이 벽만 바라보았다 - He stared blankly at the wall.
or ceaselessly:
하염없이 걷다 - walk endlessly
A formation of X-없이 would normally suggest that X would be a noun with some particular meaning, but there doesn't seem to be such a thing as a '하염'. Was there ever? if not, what's the origin of the word 하염없이?
Not exactly a reputable source, but here is some information on the etymology of the phrase "하염없이". @Mujjigun is right with the hanja - according to the source, 하염 comes from 何念(어찌할 하, 생각 념), which means 어찌할 생각 = 어떠한 생각. Therefore, 하염없이 would mean 어찌한 생각도 없이 = vacantly.
A similar phrase (but not interchangeable) is 정처 없이, which means "without a set destination." In this example, 정처 means "정해진 장소 (定處).
