Introduction and Commentary
Really cool question OP-- usually people are interested in the Black Army at the surface level but only rarely does anyone seek out information on the anarchist non-state that they sought to protect because it failed to make it through the Russian Civil War by a long shot. Lucky that we found each other though because Makhno, his movement, and broadly, the anarchism of the early twentieth century in the Russian Empire are some of my favorite subjects, so if you or anyone else has a follow-up, I'm all ears. For those interested in the Black Army more generally or their downfall at the hands of Trotsky's Red Army, please see this answer I wrote some time ago, though I imagine I'll probably rehash a good deal of what was discussed there as well.
To answer this question I'll set the stage with a short narrative here explaining how Makhnovia came into being at all and then try to sufficiently describe day-to-day life in the anarchist territory. I'll try not to skip the darker stuff by mentioning some of it here (such as Makhno's blatant terrorization of Ukraine's German Mennonite populations), but I'll likewise try not to dwell on the man himself too much since it seems you're more interested in the life of the common person on the Free Territory.
First, a note on some sources. Pyotr Arshinov wrote the absolutely enthralling History of the Makhnovist Movement (1923), which is probably about as close as you're going to get to a front-to-back history of the period from someone who watched it all play out so it's definitely worth reading, however, Arshinov was a cohort of Nestor Makhno, a convicted anarchist terrorist himself (he met Mahkno while they were both serving time and is thought to be the person who introduced the latter to formal anarchism in the first place), and about as a far from an unbiased writer as one can find considering his affiliation with Makhno personally and allegiance to anarchy theoretically. As far as it's use as a source, its Bolshevik equivalent would be an uncritical reading and citing of Lev Trotsky's The History of the Russian Revolution (1930). Anyone would agree, both contain unrivaled familiarity with the events and powerful insights into the topics being discussed, but both were written for propaganda purposes and should be understood as such.
With that in mind, I'm going to rely on some more modern sources which chronicle the Black Army's uprising, but will also pull heavily from History of the Makhnovist Movement for some of the undisputed facts in acknowledgement of its authority on the subject.
Likewise, the actual physical documentation we have concerning the Black Army (such as their proclamations and propaganda) is also on the one hand highly telling and on the other potentially controversial. To start, a great deal of it was destroyed (obviously) by the Red Army when they were rounding up and shooting the movement's leaders upon their betrayal of the Black Army so the actual physical documents themselves largely only existed outside the Soviet Union when people started studying these events. Some historians, primarily Soviet historians (as in, from the Soviet period), have therefore suggested they were altered or fabricated entirely to frame the Black Army in a better light and/or the Red Army in a worse light. Trotsky really openly loathed the anarchist movement and didn't really make any bones about that, so I think it's safe to say there are likely some issues of personal historical bias at work behind these accusations. Make of them what you will, Makhno fled Ukraine (as did many other Russian and Ukrainian anarchists) after his movement was crushed and so it's quite probable that they brought at least some of their documentation with them, but in the interest of preempting pushback of that nature, I wanted to bring it up. If one supposes, as is generally accepted, that these documents are indeed authentic then they really, really provide some additional valuable insight concerning the rather grim material circumstances of the Free Territory. Namely, a great deal of them are printed on the reverse side of other posters, old newspapers, bulk sheet paper from various factories, and similarly repurposed goods. What does that tell us? That paper was likely a scarce product in revolutionary Ukraine. Perhaps our anarchist protagonists here were just early practitioners of the three R's (reduce, reuse, recycle), or perhaps the movement was under severe pressure from all sides and facing extinction nearly the moment it came into being and therefore ran short of some of the basic necessities of a functioning state and was forced to make due with what was available to them.
The point of all this is, it's a complicated and controversial topic. Anarchism, Marxism, Revolutionary Internationalism, these are fractious and internally disharmonious movements-- especially in the early twentieth century when the basic planks of the ideologies' platforms were still being organized and assembled. Each are victims of their supposed enemies' propaganda and proponents for their own, so reading about them at the primary source level requires that some salt be liberally taken.