One notable example would be the Free Territory of Ukraine, formed in 1918 after the Ukrainian Revolution. This was an attempt to create an anarchist society, although it got caught up in the Russian Civil War of 1918-1922. While it had its own military (more or less), the area it claimed in eastern Ukraine fell under sway of the White Movement and then finally under the Red Army. Ukraine would later become the Ukrainian SSR, part of the USSR. Since it was anti-statist, calling it an actual "state" is kind of incorrect.

Another example would be Catalonia during the Spanish Civil War. Anarchists played a large role in that war and partially controlled much of Catalonia. However, it was never complete control since they had to share it with other socialist groups, militias, and trade unions (the Spanish Civil War was a real cluster fuck). This movement was obviously crushed when the leftists lost the war and Francisco Franco's forces crushed and repressed anyone opposed to him.

Yes? With a lot of qualifications though. I've written about the principal occurrences of large scale Anarchist control over regions previously, so this answer might be of interest to you. I'd be happy to expand on anything you are still wondering about after that here though, although I should point out that I'm not especially well versed on the social programs that went on in the Makhnovshchina or in Catalonia, just the military situation.

Edit: It's my own damn post, I have no qualms just reposting it here.

The Ukrainian Revolutionary Insurrectionary Army of Ukraine, or Makhnovshchina as it was under the leadership of Nestor Makhno during the Russian Civil War, is usually characterized as an anarcho-communist society. It existed for three or so years, tentatively aligned with the Reds against the Whites in the Civil War, but it was a shaky alliance at best and fighting occasionally occurred between the two, with some serious fighting happening on and off. During an off period in 1920, there was something of a formal alliance between the two as they combined forces to defeat the Whites in the region.

The campaign was a successful, but the Reds almost immediately betrayed them, arresting and executing a number of the leaders at a joint planning session late that year. Although it wasn't the end of the Makhnovshchina, the sudden betrayal broke their military strength and within a year, the Red Army had crushed them. Nestor himself hadn't been at the meeting, and led them till the end, fleeing to Paris in 1921.

Also during the Civil War was the Kronstadt rebellion, which included a large contingent of anarchists within their midst. It only lasted a week or so before being crushed by the Red Army though, so calling it a society would be a stretch. Anyways though, the same fat befell the examples you gave. Whatever the success (or lack thereof) internally, they failed because, it nothing else, they couldn't withstand the military force arrayed against them.

As we saw in the Ukraine, the Red Army betrayed the Makhnovshchina. In the case of the Commune, the French Army laid siege to the city and was willing to brutally suppress it.

In the case of Spain, it is a bit more complicated, but the end was the same. The Popular Front was a very loose association of groups, and the CNT-FAI (the main anarchist force) was quite powerful in the Catalonian region, but lacked any real allies, while the COMINTERN affiliated Communist groups were the key to foreign support from the USSR. This resulted in increased power for them, and the CNT-FAI, and non-COMINTERN groups like the POUM, becoming more marginalized, until eventually they were turned on by the Communists and fighting started to happen between groups of the Popular Front (whether that is what lost the war for the Popular Front is another issue, but it certainly did them no favors). Regardless though, the war took a decided downturn for the Loyalists with the Aragon Offensive in 1938, and the Nationalist victory obviously sealed the failure of the anarchist movement in Spain, although even if they Popular Front had won, it is up in the air whether the Anarchists could have survived given the COMINTERN domination.

Also, sometimes you see the shortlived Fiume city state of D'Annunzio called an Anarchist society. I wouldn't say that it was. Regardless though, as with the other examples, military force ended it, Italy taking control.

So in summation, the most notable attempts at Anarchist societies have all resulted from the breakdown of the previous order, and a chance for the anarchists to assert themselves. Because of this, they almost always exist in a very volatile situation, and are required to defend themselves militarily. All these examples are of societies which, due at the very least to their size, simply couldn't successfully defend themselves, and failed due to conquest.

I didn't include sources there the first time, but mainly I relied on the works of Daniel Guérin - "Anarchism: From Theory to Practice", and his anthology of Anarchist writings "No Gods, No Masters". Would recommend both - the former as a good primer, and the latter if you want to read a lot of heavy stuff.

As /u/MisterFiftyFifty pointed out, there was a serious attempt at re-making Spanish society along anarchist lines in the early months of the Spanish Civil War. After the military coup failed in July 1936, and the country spiralled into civil war, and the Second Republic collapsed. In the republican zone (the territories not held by Franco's nationalists), there was a social revolution which culminated in essentially a civil-war-within-a-civil-war.

The Spanish Revolution, which according to anarchist theorist Sam Dolgoff "came closer to realizing the ideal of the free stateless society on a vast scale than any other revolution in history", was a power struggle between traditional democrats and liberals, the revolutionary left (the socialist PSOE and communist PCE) and the anarchist unions (CNT and FAI).

Catalonia was the epicentre of the revolution, not least because it was the power base for the revolutionary left and the Spanish anarchist movement. In The Spanish Civil War, Hugh Thomas describes a "great army of nearly two million anarchist workers, chiefly in Andalusia or Barcelona, organised in the CNT (Confederación Nacional del Trabajo), and directed by a secret society, the FAI (Federación Anarquista Ibérica)." There are few other points in history where you've seen an anarchist movement of that scale.

The revolutionaries played a major role in neutralising the military rising in Barcelona, and became a significant power bloc, at least up until the May Days in Barcelona and the counter-revolution/reassertion of Popular Front control over the republican zone in mid-1937.

You had anarchist ministers in Largo Caballero's republican government, anarchist militias engaged (to varying degrees of success) on the front-lines of the civil war, and widespread collectivisation of Spanish industry and agriculture.

The anarchist were never in total control of republican Spain, and their power was substantively broken after the power struggle with the communists, but for a time there you had huge parts of Spanish social, economic and political life dominated by anarchist principles. The defeat of the Revolution is still a painful memory for many, because it was arguably the high water mark for international anarchism.

In terms of the quality of life, it's difficult to critique, because it's an intensely controversial period. You're talking about a country in complete social and political collapse. You've got an incredibly violent civil war being fought and a social revolution being waged, with multiple revolutionary movements all vying to make the new Spain in their image. So, the revolution and collectivisation in Spain is often romanticised, but there are plenty of examples of violent, coerced collectivisation and widespread killings by revolutionary groups.

There's a huge amount of excellent writing on the civil war and the revolution, but Stanley Payne's The Spanish Revolution is probably the best text I could recommend. The Spanish Civil War: Revolution and Counterrevolution by Burnett Bolloten is interesting, but has to be read with the understanding that Bolloten was (at one time at least) a communist. Anarchist historian José Peirats Valls work on this issue is also important, but should also be read with an understanding of his perspective.

The POUM and other socialist-anarchist groups in Civil-War Spain are a good example. For a glimpse of what life was like in this society you can't do better than read Homage to Catalonia, by George Orwell, who fought for the POUM. He largely focuses on his own personal military adventures as an international volunteer but does give a good outsider's overview of the general social situation at the time.

The closest I can think of is medieval Iceland. It lasted hundreds of years and was prosperous.

David Friedman has written a lot on it. An example: http://www.daviddfriedman.com/Academic/Iceland/Iceland.html