彼らは絶えずけんかばかりしていた。

English Translation

They were constantly quarreling.

絶えず and ばかり mean the same and they are redundant.

彼らは喧嘩(けんか)ばかりしていた。

彼らは喧嘩が絶えなかった。(けんかがたえなかった)

are enough and concise.

~ばかりしている/いた means “to do something so many times” or “to do it repeatedly”. 絶える as an intransitive verb means “to be extinct”, and 絶えず as an adjective means “constantly” or “eternally”.