All great periods of culture are periods of political decline. Whatever is great in a cultural sense is non-political, is even anti-political.
We can think of "the state," or the government, as politicians and their institutions; a professional class of "governors" who do the work of creating and enforcing the rules by which everyone else is expected to live. This is odd when you think about; why would one group within society be invested with the power to do this?
When I was growing up, the rationale given was that having professionals do the "political work" of society freed everyone else to "live their lives," so to speak. But we often notice that when people are obligated to act in a "political" capacity -- for example, to attend jury duty -- they are quick to defend the many obligations they have to their employer, unaccustomed as they are to deny these for any other purpose. "Living our lives" has come to take on a particular meaning in this respect!
Nietzsche puts "culture" in opposition to "politics" because where the former embodies the free creation of norms between people, the latter mandates them by force. What is "political" could be negotiated between people on their own terms; but without a specialized group "entrusted" with a monopoly of violence, such endeavors would merely be social, not political. This is what Nietzsche means when he says, "Whatever is great in a cultural sense is non-political, is even anti-political": the best things to come from human relations come freely, without one group presuming to set the terms for all others -- i.e. without power as an established principle; without "government" as an institution separate and distinct from the work of our everyday lives.
Anarchism has long recognized that what best makes a society democratic is not the "democratic credentials" of its ruling class -- not one political party that "better represents the people" than the rest -- but an end to the artificial machinery of "politics" altogether.
We can imagine life without government as entailing not the abolition of everything that government does -- like running schools or installing traffic lights -- but rather that the work of our everyday lives would change to include the many things that are now entrusted to permanent bureaucrats. But how can we take on extra responsibilities when we are already overworked as it is? The answer is: the structure of our work lives has to change. Our time would be devoted to producing what we want and need: part of the day would be devoted to the kind of economic work we do now, only its purpose would not be the enrichment of our bosses but the improvement of our communities; and the other part of the day would be comprised of relating to each other about what we want and how best to go about it. Such activity would then have the chance to develop as cultural practices amongst society as a whole.
We see that in a community of free individuals, what is "political" is superfluous; institutionalized power is only necessary when you want to preserve the rights of some against the needs of others. This is another important thing to know about the state: it has always attended the division of society into distinct and hostile classes. This is why anarchists have always identified the state as a necessary instrument of class rule.
It is important to reiterate the fact that what the state is shouldn't be conflated in every instance with what the state does. The fact that the state administers certain social programs or upholds certain laws is not proof that people wouldn't prefer similar services or "rules of the community" in a stateless society. The point is, it has to be determined what people want, and in some ways the state prevents this; but in other ways, people have forced the state to recognize what they want. So we mustn't confuse everything the state does with the underlying purpose for which it exists. Something that Marxists have been good to emphasize is the fact that the state is an important site of class conflict; for anarchists, the impulse to "smash the state" must take care not deny the importance of some things the state does which would need to be performed by communities in its absence.