。
You need it not only when you're awake, but also when you're asleep.
Literal
Eye [subject-が] is-awake time [not-only-ばかりか] is-sleeping time [at-にも] too it [subject-が] is-necessary.
~ばかりか… にも is a 'not only X but also Y' frame, with ばかりか marking the unexpected extension of X. The contrast is between being awake (目が覚めている) and being asleep (眠っている), both expressed as ongoing states with ~ている. The それ ('it') is deliberately vague — the sentence feels like a fragment of advice, with 'it' standing for whatever the broader context specifies.