There's a lot to like in the book - lots of interesting historical and anthropological observations. But I agree with Julio and Doug's reservations.
I think he overstates the case about economics' reliance on the myth of barter. I checked a few monetary economics textbooks of various vintages I had on the shelves. It is a common trope to compare monetary exchange with a hypothetical situation of barter, but not so much to claim that this was an actual historical situation - some are quite explicit about the distinction. As for whether money is essentially a creature of the state, I think Graeber is on the side of the conventional wisdom on that - and unfortunately so.
There is a lot of similarity between Graeber's argument and the opening chapter of Keynes' Treatise on Money. He draws on Keynes's favourite source on the history of money, Knapp. I was waiting for Graeber to discuss Keynes, and indeed he does admit him as an exception - "much more open to what he liked to call the 'alternative tradition' of credit and state theories than any economist of that stature... before or since"... but who "ultimately decided that the origins of money were not particularly important". (pp. 55, 57)
I think Keynes' arguments there are based on a fallacy that Graeber also seems guilty of (but I should reserve my judgement until I have read the rest) - that to find the historical origins of some social phenomenon is to have found its essence. This is quite blatant in Keynes - his potted history has an enormous gap from Hannibal's crossing of the Alps to Napoleon's, and thus skips right over the development of capitalism. Graeber gives a fuller history, but there are many points at which he implies that money's ancient history has an important bearing on what money is today.
Keynes wanted to show that money was essentially a thing of the state, so as to argue that the state could and should accept responsibility for monetary phenomena. Graeber wants to establish money as a thing of the state so as to tar it morally. There is a lot of guilt by historical association in the book, e.g., in the discussion of property and slavery in Roman law: e.g., "the notion of private property is really derived from slavery" (p. 205) (This is mixed in, though, with some good stuff about the nature of property as "arrangement between people concerning things", which does not depend at all on the historical narrative.)
Much more to say but I'll leave it there for now.
Mike