Sophocles' tragedy Oedipus the King was first staged in Athens in the early 420s BCE. In this play, Oedipus discovers that he is guilty of some of the most horrific actions imaginable to the Greek mind (or to any mind): he has killed his father and married his mother.
Regarding his punishments, one of them is self-inflicted: he blinds himself. As for other punishments, his mother/wife Jocasta kills herself. Additionally, Sophocles has hinted throughout that Laius' killer would be exiled from Thebes. Although Creon will not commit to this at the end of the play, Sophocles' audience would have known that exile would be Oedipus' punishment. Furthermore, at the end of the play, it appears that Creon is going to take Oedipus' daughters away from him.
Thus, in the course of this play, Oedipus learns that his foster father is dead, that he has murdered his natural father, that he has caused the death of his mother/wife, that he may lose his children, lose his native land, loses his kingdom, and, of course, he has lost his eyesight.
I would say that Oedipus has suffered and will suffer about as much as one human possibly could. Being put to death by the Thebans or killing himself would have been the easy way out for Oedipus. Furthermore, Sophocles' Athenian audience considered exile from one's native land a far graver punishment than we probably would today. Even Socrates rejects the possibility of going into exile from Athens and prefers to accept their penalty of death.
As for whether Oedipus' punishment fits the crime, I can't think of anyone else who has been "convicted" of killing his father and marrying his mother, so I can't point to an established precedent. The fact that 2500 years later Oedipus' name remains synonymous with these horrific actions probably does confirm that his punishment was sufficient. Like the chorus of the play, I suppose we shall have to take a "wait and see" attitude:
He was the most powerful of men.
All citizens who witnessed this man’s wealth
were envious. Now what a surging tide
of terrible disaster sweeps around him.
So while we wait to see that final day,
we cannot call a mortal being happy
before he’s passed beyond life free from pain.
(A.S. Kline translation)