Ammianus Marcellinus (late 4th century A.D.) mentions one eunuch in the imperial administration who was a good man--but says some readers are never going to believe him that a eunuch could be a good man. Evidently their reputation was pretty bad.
People always assume Byzantine or eastern Roman means Greek but it was Greek in civilizational terms. Many of the dynastic families and court officials were Armenians and the eunuch practice probably originated amongst the Armenians.