Yes.
The "no" answers seem to be trying to prove a negative about a hypothetical society based on current websites, but this is a post-disaster, alternate Earth setting we're talking about. One with its own societal conditions, and an evolved online culture. It should indeed be possible to build such a society, depending on the premise, provided we understand the question is not "can StackOverflow build up a present-day country". It is "can an online forum, e.g. StackOverflow (or any number of somewhat-different websites left over by the Old Ones, i.e. us) build up a country, defined as any possible or unforseen means of government in a realistic, future Earth setting".
The question cannot be disproved by citing the limitations of StackOverflow itself.
The "no" answers to this question suggest a fairly limited interpretation of what a "country" might look like. Sociologically, a nation-state, like a blog, is a human construct -- not a social contract or social fact, not something immutable and innate to human beings, but something intellectual that proceeds from human desire for structure, alongside numerous other, alternate means of societal organization -- temples, tribes, corporations, feudalism, etc.
By the same token, the "No" answers are predicated on an assumption that the Internet is one specific type of thing, and that other forms of organiziation on the Internet are not possible due to the nature of humanity, trolls or whatever. This is based on observation bias. The Internet would be extremely different today had the laws written in the 90s or the culture of 90s-era web communities or the intentions of people who wrote the code for the Internet been different. It is safe to assume the Internet of the post-apocalyptic future would also be extremely different. The Internet is a tool, one that mirrors offline methods of organization. The behavior of people on the Internet is a (dark) mirror of offline culture and behavior.
Both the State and online organizations are means of organizing societies from afar. In most nation states, the average citizen has little or no face-to-face relationship with those who govern them. In many ways it is a consumer relationship. To say that an online community can't run a society because then we wouldn't have the face-to-face relationship we enjoy with our local leaders, is in some ways an idealistic straw-man. It's true that such a society may be different, maybe less democratic (but not necessarily) than a town-hall democracy, but we don't live in a town-hall democracy today (and many people constantly say that technology has rendered town-hall democracy as impossible as online democracy, but we have no proof that either are impossible).
Moreover, the 30-year existence of community listservs, many of which are used to rally citizenry at the neighborhood level and the people not on the email list don't get much of a say when they show up for the town-hall meeting, proves the "Yes" argument by example.
If I were writing your story, I would focus on the concept of the "Internet of things" and how it relates to your future society both pre- and post-apocalypse. I would also read "How The Irish Saved Civilization" and its influence on so-called digital scribes who seek to preserve knowledge from data decay using the Internet (which was designed first and foremost by ARPA to preserve data in the face of a nuclear holocaust) to counteract the effects of decreasing lifespan of storage media (from books to much more fragile magnetic drives). Also keep in mind that the structure of the online community you wish to use is important. I would take the approach of sci-fi and fantasy novels that compare several different societies by having the protagonist interact with several different communities, some of which are successful, some of which are unsuccessful, some tyrannies.
You might also look at articles by respected psychologists on how social networking is rewiring the brains of young people -- not necessarily for the better, but extremely relevant to this question -- by making reputation something that carries with them throughout adulthood, where there is no break from the pressures of online reputation because they are always online, carrying it with them, and their words are not ephemeral. Also read about how China plans to use a universal "online reputation" metric to evaluate all its citizens, giving them more or less rights accordingly. At the opposite extreme, I would read about people who are heavily invested in online subcultures and their pre-existing impact on electoral politics -- the Occupy movement or the alt-right, for instance.
Twitter might be used as a model for the power of the (flash) mob, as several movements have used it as an organizing platform to overthrow governments, but also as a distraction and a sole means of communication. You probably couldn't govern by Twitter, although the current US leader will doubtless try. A society run by Twitter might appear as an "outward-facing Panopticon", an inquisitorial society where people confess their thoughts minute-by-minute, and the most influential thinkers control the discourse by virtue of who is listening to them.
Imagine a series of towns run by an Internet of Things, possibly the sole source of wisdom which survived the war (as all books have been burned up, and the written word is priceless) or people forgot how to read before the war happened and had to re-learn it, (making the online word priceless) due to a universal audio-visual culture. A town run by an anonymous message board with ephemeral posts would create a radically different society than a town run by a social network with rigidly enforced reputation heirarchy in which your comments gather you more reputation which you can exchange for credits, for instance. Imagine that currency has been replaced by credits because paper money was replaced by online credit, brick and mortar stores disappeared, manufacturing jobs were automated and disappeared before the war, and then access to credit and goods became limited to certain sites. These sites became distributed on local peers, which is how they survived the war. Only a few peers survived, resulting in a geographical focus for certain sites. Some sites are more geographically focused than others -- perhaps the people of one community do not know each other face to face, maybe never even go outside because of radiation or what-not but communicate only online -- online organizing becomes even more important. For others who know each other face-to-face it perhaps becomes oppressive, a form of social status, or a sort of techno-feudalism perhaps. Or perhaps the Internet of Things enables a small group of outlaws to maintain a shadow government which impacts the real world, but is only visible online.
Take careful note of online currencies and how those work -- including the phenomenon of "farming" goods for credit on multi-player virtual-reality sims. Certain forms of information might be power (and worth farming for credits) or online goods (and reputation) in a world with limited sources of income, where all goods of value are manufactured by machines that humans don't know how to operate, or pre-date the war.
In real life there are techno-utopians who have called for ordinary democracies to run elections, town hall meetings, citizen participation entirely online -- that is a more positive spin on it.
Any cyberpunk author would answer "yes" to this question, provided you were imaginative about it. It's an intriguing basis for a story and there's a million different ways you can go with it -- which alone disproves the "no" hypothesis -- unless you're a modernist asserting that there's only one form of ideal government -- or a postmodernist asserting that the media fundamentally distances us and prevents us from forming a community to begin with.
Really, the "no" answer to this question asserts that it is impossible to use technology to form a government, and we have no evidence of that. Or it asserts that it's impossible to modify a whole bunch of different sorts of online forums to substitute for (or run) a viable face-to-face community, again we have no evidence of that. (and some evidence that it is possible). Or it asserts that government is inherently based on a monopoly on physical force and there is no other analog or competing structure that can serve as an online model.
Certainly the existence of entirely innovative types of forum, invented relatively recently, such as the wiki (or the contacts-monitoring social-media platform) militate against the notion that it's impossible merely because you couldn't do it on an existing site, such as a Q&A site. If society were trapped underground and the only means of communication were StackOverflow, SO would quickly be adapted to serve the purpose. The question is what form it would then take.
