From the preview page of [Garnir's Dream Spaces with Hamel Bases][1], by Norbert Brunner [Arch. Math. Logik 26 (1987), 123-126]: "The situation changes completely, if one drops AC and instead assumes e.g. the axiom determinacy AD plus the principle of dependent choices DC.  Then every linear map $A:X \to Y$ from a Banach space $X$ into a normed space $Y$ is continuous."  (Brunner then says that Garnir establishes this from a weaker hypothesis.)  Also, Wikipedia tells me that the axiom of dependent choice implies countable choice, and countable choice implies that every infinite set has a countably infinite subset.  If $\ell^\infty$ (or any other infinite-dimensional Banach space $X$) has a Hamel basis, this contradicts the above setup.  A Hamel basis for an infinite-dimensional Banach space $X$ is infinite, and assuming a countably infinite subset, you can make an unbounded and therefore discontinuous linear map from $X$ to $\mathbb{R}$.

(Really this topic is not my business at all, but maybe I right as a naive student.)

  [1]: http://www.springerlink.com/content/j648651210068213/