It is well known that the space $\{0,1\}^{\kappa}$ satisfies the countable chain condition. Recall that a topological space $X$ satisfies the countable chain condition if and only if every collection $\mathcal{A}$ of pairwise disjoint open sets is countable. However, it is easy to show that the surjective image of a space satisfying the countable chain condition must also satisfy the countable chain condition, so the only possible spaces which are images of some $\{0,1\}^{\kappa}$ satisfy the countable chain condition? However, there are plenty of compact Hausdorff spaces that do not satisfy the countable chain condition such as $\beta\mathbb{N}\setminus\mathbb{N}$ or $[0,1]\times[0,1]$  with the order topology inherited from the lexicographic ordering.