- Education would be all private, resulting in an unfair class system where some people would get inferior treatment through no fault of their own.
Already the case in 1st world governments; poor inner-city kids and country kids have crappy schools, wealthy rural area kids have very well funded high quality schools.
- Healthcare -- same as above.
Ditto; free clinics, crappy insurance, and not checking issues until you have to go to the emergency room v.s. "Cadillac" insurance plans and personal healthcare concierge.
- Employment and contracts -- slavery is back! Oops, I meant "indentured servitude". Without governments getting in the way of free trade, people can be bought and sold, there's a free market for child prostitution, and owing a bad debt on a piece of paper apparently makes it all OK.
Social contracts, debts that can not be dismissed through bankruptcy (e.g. student loans), and the rest is straw man. Sure, you could buy a person in a society without government, just as you can in one with a government, but that person doesn't have to acknowledge your ownership of them, and likely no one else will, so if they leave or refuse to work for you, you're Sol. Just like it is in a society with a government.
- There is no coherent concept of 'justice'. It's loosely postulated that private justice would exist and somehow work along the lines of business arbitration. However, proponents of that nonsense seem unable to remove the issue of money and its corruptive effects from the equation.
Money is already a part of the equation in government run justice. The only difference is that this money is also used to help make unjust things legal. You can't make something unjust legal without government, so you're only left with justice and money.
- Before any issue ever reaches a (kangaroo) court, each person is burdened with the responsibility of being a judge, jury, and executioner, and perfectly rational behaviour is assumed and expected at all times. In extreme cases a misunderstanding could result in an alleged aggressor getting shot! Due to a complete lack of laws (apart from the NAP one-liner), there's no proper guidance on what constitutes aggression -- it's all hearsay and vigilantism.
That's how the legal system works now: before anything reaches a court, each person is required to go through negotiation, discovery, and be a judge and jury. Only when they fail are they allowed to take their case to court. Things like surprise evidence and witnesses that you see in the movies are extremely frowned upon. And just because you will likely get yourself shot if you found yourself in NAP society because you don't know how to behave yourself, doesn't mean everyone else will too.
- There's massive doublethink regarding the concept of 'community'. On the one hand, communities could exist as long as they abide by the NAP which is even more fundamental than any community in existence. On the other hand, the allowed communities would have no substance: all participation would fall into categories like "free trade among like-minded individuals", and voluntary participation. Due to 'self-ownership' and a rejection of competing claims on "the fruits of one's labour", communities -- as defined as "non-financial entities" -- would be impossible. A private two-party justice system would deny representation.
There is no difference between a NAP community and your local community. If you are friendly to your neighbors and expect them to be friendly to you, if you don't steal their stuff and expect them not to steal from you, if you borrow things from them and are willing to let them borrow from you, you are already practicing NAP. If, on the other hand, you currently live in a communist country, where the government owns everything, and you have and expectation that any of your neighbors are free to walk into your house and take anything of yours, and you can do the same, because everything is community owned, then I could see where you would be confused.