Choice is not needed.
EDIT: Thanks to @dfeuer for pointing out my original argument required Dependent Choice.
I will say $X$ is countable if there is an injection $f:X\to \omega$, where $\omega$ is the set of natural numbers. Uncountable means not countable.
By a continuum I shall mean a connected compact Hausdorff space.
Theorem (ZF). Every non-degenerate continuum is uncountable.
Proof. Let $X$ be a non-degenerate continuum.
For a contradiction suppose $X$ is countable. Apparently $X$ must be infinite, and so we may enumerate $X=\{x_i:i<\omega\}$ where the $x_i$'s are distinct.
Let $C_0=X$.
Suppose $n\geq 1$ and non-degenerate continua $C_0\supseteq C_1\supseteq ... \supseteq C_{n-1}$ have been defined.
Let $x^*$ be the element of $C_{n-1}$ with least subscript.
Let $x^{**}$ be the element of $C_{n-1}$ with least subscript greater than $x^*$'s.
Let $\mathcal U_n=\{U\subseteq X:U \text{ is open, }x^*\in U\text{, and }x^{**}\notin \overline U\}.$ Since $X$ is Hausdorff, $\mathcal U_n\neq\varnothing$. Let $$\mathcal C_n=\{C\subseteq C_{n-1}:x^{**}\in C,\;C\text{ is connected, and }(\exists U\in \mathcal U_n)(C\cap U=\varnothing)\}.$$Let $C_n=\overline{\bigcup \mathcal C_n}$. Then $C_n$ is a continuum, and is non-degenerate because some elements of $\mathcal C_n$ are non-degenerate. This is true because compactness and normality of $X$ implies the quasi-component of $x^{**}$ in $C_{n-1}\setminus U$, $U\in \mathcal U_n$, is connected, and this quasi-component must meet $\partial U$ in order for $X$ to be connected. (Choice is not needed to prove normality, nor is it needed to prove these quasi-components are connected.)
Continuing in this manner, we construct a nested sequence $(C_n)$ of non-empty compact sets. Their intersection must be non-empty. But on the other hand we ensured each point of $X$ is eventually not in $C_n$. Contradiction. $\blacksquare$
Follow-up question: Is every separable metric continuum equinumerable with the reals, in ZF?