"Anarchist state" is a contradiction in terms but certain geographical regions have experienced a degree of anarchy or anarchistic social structures in recent history.
Southeastern Spain during the mid-1930s is often brought up as an example, maybe most famously by George Orwell (writer of 1984 and Animal Farm) when he traveled to Spain to fight in a socialist militia and later wrote about his experience and particularly his admiration for the anarchist-controlled region of Catalonia.
The Free Territory that existed primarily in southeastern Ukraine during roughly 1918-1921 tends to get brought up.
There was an anarchist region somewhere in Korea during 1929-1932.
The Mexican Revolution had significant anarchist elements (often referred to as Zapatistas, Magonistas, or rarely, "liberals" of a certain variety) and there were some anarchist strongholds that held out against the end of the revolution.
And of course there's little to point either way to what extent anarchy might have existed in any number of ancient societies with no recorded history, or a history recorded and interpreted by people with their own ideological biases. Men in caves.
And-and, as an important sidenote to this, framing the question in terms of "society" doesn't necessarily specify the scale you're talking about. Are anarchist neighborhoods included here? Anarchist cities? Packs of anarchistic nomads with no set geographical region to claim? If anarchistic relationships and interactions happen within a society that might otherwise not be anarchist, does this count as becoming closer to an anarchist "society" (or a destruction of society as we know it)?
This is a really old question so I doubt these semi-rhetorical counterpoints are going to get answered, but looking at this particular Q&A didn't seem satisfactory to me.
