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Whenever one employs the term "equality", one must also define if one thereby means equality of opportunity or equality of outcome. The more your society has of one, the less it will, in general, have of the other.
I assume you mean equality of outcome, given the context.
It is true that under anarchy, there would be inequality of outcome. But there would be a very high equality of opportunity. Contrast this with the non-anarchic, e.g. democratic, societies that we know from our time. Here, those who strive to provide the most value to others are taxed heavily, while those who do not provide value, are taxed lightly. This creates some equality of outcome - although extremely inefficiently due to bureaucracy and so on - at the expense of equality of opportunity.
Under anarchy, corporations are not, in fact, free to exploit. They do that today, under democracy. That is because democratic law allows for the existence of corporations and they are able to game the system, for example by heavily influencing the laws that are passed.
Under anarchy, there is no government and thus no monopoly of violence. The entire apparatus whereby corporations today exploit the masses (take for example the "too big to fail" bailouts) is completely non existent under anarchy.
Anarchy does specifically not rely on people to want what is best for society. In fact, socialism does. This is why we have numerous records of failed socialist societies - the U.S.S.R, Venezuela, etc - while the societies coming closest to anarchy have been very stable (google Zomia).
Under anarchy, people will enter into voluntary agreements for protection from harm by others, leading to effective and mostly non-violent policing by private security providers. Unlike the police we know under democracy, these will have every incentive to protect their customers and employees.
Even greedy people are, under anarchy, only able to obtain the objects of their desire through voluntary means. They will have to produce something of value to others to become rich. While under socialism, you only need to rise up in the party and you will have the power to steal and rob - that is, obtaining wealth by removing wealth from others.
Health care and other vital services would obviously be well provided under anarchy: everyone needs them, so there would be high demand and sharp competition to provide the best services most effectively. Again, this is different from what we see under democracy - so called "public" health care is extremely inefficient and uncaring, often leading to people dying simply because this has no consequences for anyone.
So anarchy is in fact the best way to secure robust and beneficial social structure including a network of vital services.
I am probably a good person but I haven't taken the time to fill out my profile, so you'll never know!