My knowledge of anarchism is based on decades old readings of mostly Proudhon, Bakounine and Ivan Illich. Sorry I can't give you precise references. Note also that none of the following implicates endorsement.
Actually reading anarchists thinkers it is obvious that institutions would still exist in the world as they envision it. No one advocates for the disappearance of humans gathering their effort into collective projects, just radically changing the organization of those, including anarcho capitalists, who are the only type of right wing anarchists I know of. The difference is left wing anarchists seek to diffuse as much as possible the distribution of power in those associations, while anarcho capitalists rely purely on individual property (so schematically, in one case the workers rule the factory, hospital, school etc, and in the other it is the owner(s)).
Left anarchists, notably Proudhon, propose associations that could be comparable in size to the nations states we have today, but organized along three main principles:
- subsidiarity, it is to say that authority on decisions is distributed bottom up and not top down. Instead of the topmost government authority (president, congres, etc...) deciding what lower echelons (city councils, neighboor associations, etc) can figure on their own, the lower freely decides what topics need to be decided by upper levels. For exemple, the city council decides that roads would be better managed by a federation of cities in the area and send delegates to figure it out, what Proudhon called a federation of communes. It is somewhat similar to the states right VS federal power controversy in the US.
- self-management: Nobody holds a position of power unless they have been specifically approved by the whole community to fit this role. Important decisions are taken during general assembly gathering all members of the community. Since some decisions need to be taken timely (what surgery to perform on this wounded person?) or whould be overwhelming to decide collegially (is there pasta on the menu today ?) delegates are designated to handle those. In order to avoid abuse of this power they receive specific mandates (they can't take decisions they were not expressly authorized to take), are revocable, and must be strictly transparent. This could include institutions usually considered antithetic with anarchism, like police or the army. A community might need people trained to handle misbehaviour or repel aggression. Simply, an anarchist policeman is bound to be elected to their role and to be transparent about it (for exemple, interrogated after any violent act to make sure it was necessary and proportional).
- free association: nobody is bound to a community like a citizen to a state. If one does not like the rules in a place - although they participated to their making just as much as any other member - they can leave. This has obviously difficult implications when it comes to a neighborhood or a town, rather than a factory or a farm. I dont know how anarchists solve those contradictions.
So under those principles one can imagine how a factory, a school or an hospital could be organized. A school, for exemple, would be run by its teachers and students (provided it is practical, toddlers ruling themselves seems an overstretch). Self managed schools do exist where students can freely chose to attend classes, programs are debated during assemblies and menial tasks like cleaning or cooking are shared. Programs could be delegated to the federation level if the members of the school think it can be beneficial (standardizing education has merit after all). The main point being that students are taught autonomy as a group rather than obedience to rules decided by others who are not submitted to them like in most schools. Ivan Illich, specifically, worked on the subject of autonomous schools.
An hospital would be run by every member of the staff, from surgeons to orderlies, and possibly patients (who are a part of the community after all). Life or death decisions that need to be taken in a split second would be delegated to surgeons, when the total amount of material and drugs they can be allocated would be decided during the assembly.
Worker cooperatives are also an actual exemple of institutions based on anarchist principles.