No.

The Free Territory was not a socialist project so much as it was a territory under the protection of socialists. The Black Army's primary objective was defending from German, Austrian, and White Army incursions. Its destruction by another group of socialists with different political priorities -- the Red Army -- makes for a great "What if?" story, but this lost cause narrative obscures the realities of what anarchism in Ukraine actually looked like.

Historian Colin Darch writes in "The Myth of Nestor Makhno" that only a very small portion of people in the Free Territory lived in anarchist communes. Only five of these communes existed within the Free Territory containing over a million people, and the first was destroyed by Austrian soldiers after a few weeks. None of them contained more than a few hundred inhabitants. The diaries of its inhabitants, and Makhno himself, indicate that the commune was basically a haphazard military camp, and not particularly socialist:

There is nothing on social relations of production, on the division of labor, on crop selection, on the labor process, on marketing, on the distribution of surplus; simply three hundred undifferentiated anarchists and peasants in a communal canteen, taking a day off whenever they felt like it. And these few weeks in spring were to serve as a basis for a social revolution.

Within Free Territory they controlled, most of the changes the Makhnovists did were more martial than socialist. They enacted monetary policy, they regulated the press, and used military force to enforce these new rules (Alexander Skirda, "Anarchy's Cossack"). Broadly, nothing about the rule of the Makhnovist Black Army was substantially different from life under other occupying forces during the Russian Civil War, except that the Black Army (as well as the Red Army) were far less likely to massacre Jewish populations than the Whites.