Their point isn't that you don't need anything, but extremely little in comparison to the desires that we feel. See AO27, speaking of the poor: 'On the contrary, such people know
that they are close to grass, almost akin to it, and that desire "needs" very few things...'

The reason for the apostrophes is that even those things, however, you don't need because you lack them: 'what is missing is not things a subject feels the lack of somewhere deep down inside himself, but rather the objectivity of man, the objective being of man, for whom to desire is to produce, to produce within the realm of the real.' (Ibid)

Why isn't starvation lack? Deleuze explains this pretty well in his lectures on Spinoza. A lack is always based on the representation of the integral whole. So a one-armed man (Spinoza's example) doesn't lack a second arm until he imagines an essence of the human being in relation that he does not correspond to. Similarly, an anorexic, no matter how hungry they are (and this is certainly no effort to defend anorexia as a mindset) do not 'lack' food, because they no longer have the representation of being full as a normal state.

To do away with lack therefore means that there is no longer any baseline or equilibrium in relation to which these desires are thought. You can still eat, obviously, but you eat in order to produce something positive out of your malnourished body, not in order to restore a lost normal state of being full.