"I am not allowed to eat."
Translation:Απαγορεύεται να τρώω.
Απαγορεύεται is an impersonal verb. The phrase literally means "it is forbidden that I eat", where it has been dropped from the Greek sentence, and that I eat/να τρώω is the verb's subject.
I'd say that "I am not allowed to eat" would be translated a bit more accurately as "Δεν (μου) επιτρέπεται να τρώω". Επιτρέπεται is still impersonal, but at least the sentence keeps the allow and the negation.
No, με is needed for a direct object, and the verb here is passive, so it has no object. Μου, on the other hand, is used for indirect objects (for me, to me etc), as in this sentence.
Let's break this sentence down:
Απαγορεύεται = It is forbidden
να τρώω = (me) to eat
μου= to me
Μου απαγορεύεται να τρώω = (lit.) It is forbidden to me to eat = I am not allowed to eat.