You would think that the theme of exile in Libation Bearers would center around one figure: Orestes. After all, it's Orestes who is coming home after many years abroad to kill his father's killers and reclaim his birthright. In fact, all of the play's "good guys" are exiles of one form or another. Electra considers herself an exile because her father is dead and because she feels alienated in her own household – she can't cozy up to mommy Clytemnestra, her father's killer, much less to Aegisthus, Clytemnestra's love interest. The Chorus, meanwhile, are exiles in the sense that they come from other cities that were defeated in war; now, they serve the royal court of Argos as slaves. As for Pylades, he certainly isn't an exile in any strict sense, but he is far from his home, risking his neck on behalf of his buddy Orestes.
This sense of rootlessness in the main characters contributes to the play's overall mood of tentative hope – Orestes, Electra, and the Chorus, at least, seem to think that killing Agamemnon's killers will make them more at home in Argos. For Orestes at least, these hopes are shattered at the end of the play. Killing his mother leaves him emotionally and mentally destitute, and he runs off to Delphi for purification. Orestes is even more homeless and outcast at the end of the play than the beginning.
At the end of the play, every character remains an "exile" in the same way that he or she was at the beginning of the play.
For the characters in the play, being close to one's parents is a central part of the idea of home. Thus, when one is orphaned, one becomes an "exile" by default.