This answer is dipping heavily into my lay-interest in anthropology. Because of that I want to preface it by saying that, as fascinating as I find anthropology to be, it is a pretty awful field of scholarship, with a terrible history of serving the interests of settler and colonialist interests in subjugating the very people it studies. Many modern anthropologists try to avoid this, but it is a thing, and there is a deep and deserved skepticism of anthropology on the part of colonized people, not least because it most often takes the form of the colonizer using their research to define the lives of the colonized. Fuck that noise. Because of this, I suggest scoping out the following in addition to the stuff I mention below:
- Custer Died for Your Sins by Vine Deloria Jr. (or Anthropologists and Indians, which is a pamphlet from Little Black Cart that includes probably the most relevant essay and other related material)
- Decolonizing Methodologies: Research and Indigenous Peoples by Linda Tuhiwai-Smith, which is pretty academic and is geared towards both researchers of indigenous peoples and indigenous peoples who might be the subjects of said research, but includes a pretty thorough examination of how research often harms the researched.
Many ancient cultures (and some current cultures) are organized along lines that are not centered on state. In fact most non-agricultural peoples would have what I consider to be non-state social formations (hunter-gatherers, many horticultural and herding societies, and even some agricultural societies), the state being a particular form of hegemonic social order (of course, different people define the state differently, here is a link to an old question that perhaps we should be revisiting: http://www.anarchy101.org/9913/can-someone-please-explain-the-state-in-simpler-terms.
While there are lots of cultures that have lived without the state, that doesn't mean they were perfect or a model of what anarchists necessarily desire. Many still had/have social hierarchies that anarchists would reject. Many were or are patriarchal, or have very strict gender roles and norms (though there are also many that are less rigid in their gender roles and have less gendered social stratification than the majority of state societies). To my thinking, being anarchist is not just being against the state, or just the state and capitalism. There are many many more spooks we need to chase away.
Regarding starting places for research, baa, fa, and human offered some really good suggestions. I would also suggest checking out Society Against the State by the late anarchist anthropologist Pierre Clastres, who studied the Guarani, Yanomami and Chulupi in South America.
While I am less a fan, Harold Barclay's work might be of interest as well, specifically his books People Without Government: an Anarchist Anthropology, Culture and Anarchism, and The State. He is also both an anarchist and an anthropologist, and his anthropological work was primarily in North Africa (Egypt and, I believe, Tunisia).
I would also highly suggest Stanley Diamond's book In Search of the Primitive, and, for an interesting take on state formation, Fredy Perlman's Against His-Story, Against Leviathan!