The idiom 자나 깨나 (meaning "whenever one sleeps or wakes up") is used to say always.
자- is the stem of the verb 자다; 깨-, the stem of the verb 깨다. To use these verbs, you should attach an ending (or multiple endings) to each stem. Regarding the idiom, the ending -나 (or -으나) is defined as follows:
여러 가지 중에서 어느 것을 선택해도 상관이 없음을 나타내는 연결 어미.
A connective ending used when one's choice does not matter despite many options.
For an idiom, the meaning of every word does not remain exactly the same, and part of the idiom may not be logically explainable. You may have thought that you cannot wake up of your own free will. Regardless of whether it is possible to do so, however, 깨다 is no more than an antonym of 자다 in the idiom. There is another idiom which is semantically the same as 자나 깨나: 앉으나 서나. 앉다 ("to sit") and 서다 ("to stand") are also antonyms. Sleeping, waking up, sitting, and standing are not the only actions that you do in a day, but the idioms mean always. The same logic applies to an English idiom, "rain or shine," although it is a bit different from 자나 깨나. Anyway, these three sentences mean the same: