I should point out first of all that it's wrong to say "the Athenians hated Sparta". There were periods when Athens was at war with Sparta, but there were also significant periods when Athens and Sparta were allies (most notably during the Persian Wars, but also against Thebes in the 360s BC). During these times Athens proved itself a committed comrade-in-arms. Moreover, there wasn't just one opinion in Athens; even at times when Sparta was an enemy, there were plenty of people in Athens who admired the Spartans and wished things could be different.

This is important because the favourable view of Sparta - lakonophilia - is the origin of many of the relevant Athenian sources. These texts weren't written to slander Sparta, but to glorify it. They reflected an admiration for Spartan laws and customs and a sense among the Athenian elite that things would be better if Athens were more like Sparta.

This makes sense when we bear in mind that the different forms of government that existed in Classical Greece served the interests of different groups, and the group that produced practically all of the surviving literary sources - rich old citizen men - would be far better served by a Spartan-style oligarchy than an Athenian-style democracy. In Sparta, only rich men could vote; only rich citizens above age 60 could be voted into the gerousia, the oligarchic council that set the agenda for Spartan policy. The Spartan elite didn't have to worry about persuading the masses or about the petty interests of small farmers, urban shopkeepers and rowdy sailors. They sat among their peers and decided on matters of state as they saw fit. There were plenty of men in Athens who thought this sounded pretty great.

We should also bear in mind that some of our most important information about Sparta comes from men the Athenian democracy had spat out. Thucydides learned most about Sparta after the Athenians exiled him for his failures as a general in 424 BC, giving him the freedom to travel the Peloponnese and talk to Spartan commanders with a credible story to show that he wasn't a spy. Xenophon was probably exiled precisely because he offered his services to Sparta as a mercenary. Cast out by his home country, he was embraced by the Spartans and their king Agesilaos, and spent more than 20 years living in the Peloponnesian countryside on an estate they granted him. Why should these authors hate Sparta, the country that did them more favours than Athens ever did?

Now, I don't mean to just play the contrarian here. It wouldn't be true to say that our Athenian sources loved Sparta. With the exception of outright laconophile screeds like the fragmentary works of the oligarch Kritias, they tended to offer a mixture of praise and criticism. Plato, for instance, describes many Spartan practices as exemplary, but his ideal constitution is not simply a carbon copy of Sparta's. Aristotle (though not an Athenian) similarly saw much in Sparta that he admired, but also criticised their laws, especially when it came to the rights of women. Xenophon clearly thought highly of the Spartan upbringing (going so far as to commit his sons to it) and the Spartan military system, but he also made it clear that he thought of the Sparta of his own day as a callous, hubristic, degenerate version of its supposedly glorious past. Isokrates was happy to present Sparta as a model democracy (if you will believe it) and a model of military organisation, but he was perfectly aware that the state was past its prime and no longer able to take a leading role in Greek affairs.

You'll notice that the things these authors don't like about Sparta have nothing to do with the Spartans being "murderous pedophiliac slave drivers". Those aspects of Spartan society were not only acceptable and normal in their eyes, but positively praiseworthy. The murder element, insofar as it applied to enslaved people and enemies in war, was uncontroversial; the pederastic practices of the Spartans are repeatedly and explicitly praised by Xenophon (though he pretended they were not physical relationships); as for the enslavement, it would be extremely hypocritical for Athenians, of all people, to criticise the Spartans for that. While the number of enslaved people in any Greek state is a matter of sheer guesswork, it's likely that there were as many of them in Athens as in Sparta, if not more. Athens was a slave society, in the sense that it relied on enslaved people to make its entire way of life possible. The only comment we get from Athenian authors about enslaved people in Sparta is that their typical dress makes it safer to beat them if they get in your way in the street, since you'll always be sure you're not beating up a free man by mistake.

In short if we now think of the Spartans as pretty horrible people, it is not because our Athenian sources are deliberately making them look bad. In fact the opposite is true: they mostly try to polish the Spartan turd (and this gets much worse after the Classical period, when Sparta ceases to matter as a political force but grows into a huge cultural-philosophical icon). Many Athenians admired (parts of) Spartan society and wanted others to think of it as an example to follow. If we nevertheless get the impression that Sparta was a bad place full of awful human beings, we are already reading between the lines.

 

Incidentally, it is clearly not true that the Spartans kept no records at all. During the Roman period, the Boiotian author Plutarch was still able to travel to Sparta to consult written laws and genealogies. The so-called Great Rhetra he cites in his Life of Lykourgos is generally assumed on the basis of its Archaic language to be a genuine preserved text, perhaps 700 years old at the time Plutarch read it. It is likely that such things were initially preserved as oral tradition but were eventually written down as Sparta became more concerned - like other states - to codify its laws for future reference.

Iphikrates has made a great post so I won’t cover much of the same things: but just to add that this idea of the Athenians warping our perceptions of Sparta became such a contentious historical issue that is has a term still used by Historians: “the Spartan Mirage” first penned by Francois Ollier in the early 20th century. As the name suggests historians began to suspect whether the Sparta presented to us by the Athenian historians is merely a warped fabrication based on their own cultural and political prejudices.

As the other post pointed out many Athenians such as Xenophon and Plato greatly admired the system and as such are considered Laconophiles: “lovers of Sparta”. Xenophon was a close friend of the Spartan King Agesilaos, lived in Olympia for a time under Spartan approval and some have suggested his own sons underwent the Agoge: Sparta’s harsh training system, so we know that he at the very least had a close connection with the Spartans and would be able to observe their customs and practices first-hand, even if he does overly-praise them. As the other post mentioned others such as Thucydides would have been able to gather reports and speak to Spartans whilst in exile during the Peloponnesian War. Plutarch corroborated contemporary reports and as Iphikrates pointed out was able to use some of the (few) written records left by Sparta such as their Rhetras or Laws. There are also fragments of Spartan poetry from Spartans such as Tyrtaeus and Alkman that give us an insight into Spartan life and in particular their values and worldview so they are invaluable for historians on ancient Sparta.

I would just like to clarify something though which was mentioned: though many Athenians greatly admired the Spartan system it was not solely because they resented the democracy and wished for power to be in the hands of the minority. Sparta wasn’t strictly just an oligarchy: the Spartans had a popular assembly called the Apella whereby any Spartan over the age of 30 could attend and vote on laws and policies, then voted in the annual magistrates including the most powerful magistrates in Sparta: the Ephors and the Gerousia. Furthermore Ephors and Gerousia members were selected from among the general populace, Aristotle says the Gerousia were selected from the “best” men which some have interpreted as the richest, but this isn’t definitive as Plutarch and Xenophon do not mention this but talk about them being the most noble and virtuous. Either way Sparta wasn’t strictly just an oligarchy but was a unique blended constitution that mixed elements of different systems.

What a lot of the Athenian observers praised about the Spartans was their internal stability, characterised by the Greek concept of “eunomia” which means “good order” or harmony. They looked at other systems (such as Athenian democracy in the Peloponnesian War) and saw the volatility of the masses when making decisions in a time of crisis, they then despaired at how in other cities the oligarchs played a zero-sum game and acted in their own self interests which caused internal strife, and of course resented monarchy and tyranny for limiting the freedom of all citizens. So they looked at Sparta with its blended constitution (Kings: Monarchy, Gerousia: Oligarchy, Assembly: Democracy, Ephorate: combination of the others) and saw a great degree of internal strength that contrasted with Athenian volatility.

Furthermore not all Athenians are pro-Sparta. Aristotle (not strictly Athenian but he did live there for a time and was tutored by Plato) condemned the system and points out all of its flaws if you read his works. So as historians we can strike a balance between the very favourable accounts of Xenophon and Plutarch with the critical account of Aristotle.

On top of this there has been some archaeology in Sparta, though the evidence is rather fragmentary and there isn’t much left what we do have matches a lot of the written records.

We can never be 100% certain, but with what is left we are able to piece together a fairly coherent picture of Spartan society

Edit: If you’re interested in perceptions of Sparta and the Spartan Mirage there’s some info here to get you started: http://blog.yalebooks.com/2016/10/06/was-there-a-spartan-mirage/