What kind of anarchism are we talking here?

A nation-state has three tasks to fulfill:
Security: Both inside and outside, and no lots of people with guns cannot defend against proper armed forces with air-superiority or even against terror-bombing and worse things. I would argue that inside security could be replaced by enough guns, or a lack thereof. But this comes down to your opinion about mankind, as humans are vastly different saying there are no violent humans is very likely more wrong than right though.

Welfare: A state also has to make sure the weakest can not become too weak and as the USA very clearly show, private donations are far from enough to keep the people well and make sure that as many people as possible can live a good life. In strong contrast to countries like Sweden, Denmark or Switzerland, which even hands out free drugs to addicts, which in turn has a net-positive effects by saving money, compared to letting them die ( which would also create social-alienation for the family and friends ). Vast social-security and free psychological assistance makes sure social-cohesion stays relatively strong.

Last but not least, Freedom:
Yes, the state has to protect your freedom both in and outside, this overlaops with security. But also your freedom to engage in contracts, contracts can only be upheld via violence and unless you were ready to execute violence on others with all means necessary you'd be unable to have economic freedom, when it comes to binding contracts. Of course some people would prefer a world without corporations, but as clans and gangs are corporations as well ( corporation is a social-organization style which encompasses far more than incorporated business ) it is likely that they become governing bodies, as those that feel socially-alienated or even rejected will search for their own group.

I would like to hear your rebuttal to these specific points.