"자기로 돼 있다" You need to understand 되다 and 돼 있다(되어 있다 = 되다 + -아/어 있다) to understand such a phrase. - 되다 = 1) become, turn into (something), 2) come to, get to (a certain state or situation). - -로 돼 있다 = is set to; is arranged/scheduled to (do or be something). Examples: - 일본에서 활동하다. (be active in Japan - actual activity) - 일본에서 활동하게 되다. (come to be active in Japan - emphasizes the circumstance of coming to the outcome) - 일본에서 활동하기로 되다. (is set to go active in Japan - emphasis is on the process of deciding on it) - 일본에서 활동하기로 돼 있다. (is in the waiting to go active ... - greater emphasis on the state of having decided and is now in the waiting/preparation) With the above in mind, 자기로 돼 있다 sounds a little odd, since sleeping(자다) is not something serious or complicated enough to be scheduled or arranged. It might mean "arranged/scheduled to sleep with someone (with a sexual connotation)", which might make a bit more sense but is still awkward. So it doesn't appear to be a phrase you would commonly hear or say.

I think I should know the context where it is used first... but I guess it means "I'm supposed to sleep"