Who is the narrator, can she or he read minds, and, more importantly, can we trust her or him?
Third Person (Objective)
Technically speaking, The Eumenides doesn't have a narrator at all, because it's a play. Instead of hearing what characters do, we actually see them do it. Of course, these characters also talk about themselves, in which case they briefly become first-person narrators, and also about each other, in which they case they briefly become third-person narrators.
All the same, because these are just isolated instances within a larger structure, it makes most sense for us to think of the play as having a Third Person (Objective) perspective.