Having inspected the proof of the fact that for every compact metric space $X$ there exists a continuous surjection from $\{0,1\}^{\mathbb N}$ onto $X$, it looks to me it is a theorem of ZF + DC (dependent choice).
Is there any reference for this fact in the literature?
The proof I would use, is to use that $X$ has a countable base (this needs countable choice, I think), and so $X$ embeds into $[0,1]^\omega$, quite explicitly. And as the Cantor function $c: C \to [0,1]$ maps $C$ onto $[0,1]$ (also choicelessly), $C$ maps onto $[0,1]^\omega$ too, with $c'=c^\omega$ essentially. Then Kechris in his book shows that any closed subset of $C$ is a retract of it (using branches in a finitary tree argument, I recall it as being quite explicit), so $c'^{-1}[X]$ is too, and hence $X$ is an image of $C$ (composing the retract with $c'$). Cursorily I only see the Urysohn embedding (from a countable base) as the part that would need countable choice.