It is important to remember anarchy is not only an economic system, but a political one as well (or lack thereof). It demands the absence of a monopolistic entity which exerts dominance over society through force, as opposed to competitive means.
It is entirely possible to live in a world without centralized rulership, while still having rules and laws governing human interaction. Organization would happen on a voluntary basis, and those who break the law would still be punished either directly or indirectly through economic ostracization.
Anarchy is not an economic system, it is a social system. Anarchism is based, in part, on the moral premise that we should not force each other to do things,and that no one has the authority to rule over others.
Communism does not require enforcement of property "rights", and state welfare is unnecessary. In a communist society, there is no incentive for anti-social behaviour, and murderers would either be helped (to socially acceptable behaviour) or ostracized by the community. No military-no need for WMD.
Several anarchic movement (Spanish Revolution of 1936, Makhno movement in Ukraine, the start of russian revolution of 1917) was quite economical successful in the past. The reason why they failed was mainly military reasons.