"Eleni has the book."
Translation:Η Ελένη έχει το βιβλίο.
The rule is that proper nouns always need an article as subjects or objects of a sentence because it sounds weird otherwise. Not using it is equivalent to saying in English "car has broken down". The only case I can think of that you can use a proper name without an article is when it is a predicate in the sentence, as in "Το όνομά μου είναι Ελένη", meaning "My name is Eleni".