The short answer is that there is at least a thousand years between the height of ancient Spartan military power (~500 BCE) and however you want to define the height of Byzantine (medieval Greek/"Greek")^* civilization (~500 CE if you're going for Justinian or ~1000 CE for Basil II). The enormous difference between the two points in time isn't really due to some kind of decline, military or otherwise, brought on by the introduction of eunuchs (eunuchs could command armies in the Byzantine period, and even quite successfully) but simply due to the huge span of time between them.

Castration is a very old institution (especially among Greece's neighbors in the ancient Near East), but in the ancient period eunuchs were not nearly as popular in Greece as they were in, say, Persia. The Greco-Roman world had a lot of predictable negative stereotypes about eunuchs: they would seduce your women (or your men), they were an indeterminate, artificial category between men and women (Aristotle, for example, seems not to talk about them often for exactly this reason), they were cowardly, gluttonous, and so on. Eunuchs might have been slaves, or, later into the Roman period, freedmen; those eunuchs that would have been present in classical Greek society might be non-Greeks (though not always). There might have been a small population of eunuchs in a very cosmopolitan city like Athens at the time of the Peloponnesian War, but I don't think I've ever heard of eunuchs in Sparta. Once the Hellenistic period begins eunuchs seem to play a slightly larger social role - the Ptolemies in Egypt seem to have employed a lot of them, and this is also when we start to see the rise of certain cults (like that of Attis) involving castration. Even once Greece is conquered by Rome, under the early Empire (1st c. CE) eunuchs seem to be acquiring more roles in court circles: Domitian's (possible?) boyfriend Earinus was a eunuch, and is mentioned by several Latin poets (one of whom complains of the difficulty fitting Earinus' Greek name into Latin poems). By the time of Constantine, eunuchs are pretty firmly entrenched in the population of the imperial court; Julian the Apostate (r. 361-363) tries to get rid of them, but as soon as he dies they come right back and stay more or less for good - at least until Alexios I Komnenos at the end of the 11th century (though even he doesn't get rid of eunuchs completely).

Where are the Byzantines getting all these eunuchs? For a while eunuchs were imported from outside the empire - Narses, one of the most famous Byzantine eunuchs and one of the best Byzantine generals, was from Armenia - but around the 9-10th centuries we start to see more eunuchs originating from within the empire's borders, usually from Paphlagonia in northwestern Asia Minor. I've heard that there were also certain barbarian areas in western Europe that supplied Byzantium with eunuchs, but I can't think for certain of a citation on this. This doesn't mean there were never any foreign eunuchs in Byzantium after this point - an Italian bishop on an embassy to Nikephoros II brought him a gift of four eunuchs, who had possibly been purchased in Spain - but by this point the Byzantine demand for eunuchs seems to have been met by internal supply. Castration wasn't even necessarily thought of as a bad fate: aside from great potential spiritual benefits (perfect, effortless chastity was often cited), it offered a kind of social mobility, since any eunuch (no matter where they came from, inside or outside of the empire, or their family/economic background) could theoretically achieve very high office in the imperial bureaucracy.

Outside of Byzantium, by the High Middle Ages castration almost always seems to be a punishment of some sort. (Though this is also true within Byzantium - deposed claimants to the imperial throne could be castrated, as well as I believe pedophiles and homosexuals.) Abelard (a huge celebrity of his time - basically the David Bowie of Medieval Christian Scholasticism) is castrated after his relationship with his student Heloise gets discovered by Heloise's uncle; in the Netherlands, it was one of several possible punishments for bestiality. Westerners who went to Constantinople (like the 10th-century bishop Liutprand of Cremona) seem mostly perplexed by eunuchs: they definitely knew the Byzantines valued them (Liutprand records a nasty joke made by a commander who recaptured a Byzantine fort in South Italy: regarding the punishment of the garrison who resisted him, "since your emperor loves eunuchs so much, I'll send him some more"), but they seem very uncomfortable dealing with eunuchs themselves (Liutprand complains that the eunuch bishops he keeps meeting in Byzantium went against western understanding of canon law).

I can't really speak for anywhere outside of Byzantium, but I hope that answered at least part of your questions! If you'd like some further reading on anything I mentioned, please don't hesitate to ask.

---

^*) Whether the Byzantines were Roman or Greek (or something else entirely) is a hotly-contested subject even today. Personally I think they're Romans, but I'm just a grad student, not really an authority on the subject of Byzantine identity. I can provide more sources (including a very recent book) on this if you're interested - just let me know.