Suppose that $(X,2^X)$ is equipped with a non-atomic probability measure $\mu$ (the existence of such spaces is consistent with ZFC). This induces the $L\_1$ pseudometric $\Delta$ on $2^X$, via 
$\Delta(A,B)=\mu((A\setminus B)\cup(B\setminus A)$.


Question: Assuming ZFC, does the (pseudo)metric space $(2^X,\Delta)$ contain an $\epsilon$-separated subset of cardinality at least $|X|$, for some $\epsilon>0$?


A subset $N$ of a metric space $(\Omega,\rho)$ is $\epsilon$-separated if for all $x,y\in N$ we have $\rho(x,y)>\epsilon$.



(non) separability of the power set
The question has a trivial negative answer, as long as an atomlessly measurable cardinal exists (if one doesn't it is vacuously true, of course). Given an atomless probability measure $\mu$ on $(Y, \mathcal{P}(Y))$, let $\lambda$ be the supremum of the cardinalities of the $\epsilon$-separated subsets of $\mathcal{P}(Y)$. We can then define $X = Y + \lambda^{+}$ and extend $\mu$ to $X$ by defining each subset of $\lambda^{+}$ to have measure 0. This is an atomless measure on $\mathcal{P}(X)$, and any $\epsilon$-separated subset of $\mathcal{P}(X)$ defines an $\epsilon$-separated subset of $Y$ (by intersecting with $Y$, which doesn't change the measure). So there is no $\epsilon$-separated subset of $\mathcal{P}(X)$ of cardinality $|X|$, because $|X| \geq \lambda^+ > \lambda$. 


So I will reinterpret the question to give a more interesting positive answer, along the lines of what Ashutosh suggested, which is to say, $X$ will have atomlessly measurable cardinality $\kappa$, and $\mu : \mathcal{P}(X) \rightarrow [0,1]$ will be the atomless $\kappa$-additive measure witnessing this fact. The material involved is adequately treated in [Fremlin's textbook](https://www1.essex.ac.uk/maths/people/fremlin/mtcont.htm), though due to its size it is important to know where to look.


For any probability measure space $(X,\Sigma,\mu)$ we can form its *measure algebra*, $A = \Sigma/\mathcal{N}(\mu)$, where $\mathcal{N}(\mu)$ is the $\sigma$-ideal of sets $S \in \Sigma$ such that $\mu(S) = 0$. That is to say, in $A$ we identify measurable sets up to null sets. For probability spaces, $A$ is a complete Boolean algebra.


The pseudometric $\Delta(A,B)$ in the question then becomes a metric on $A$, which Fremlin says is called the *measure metric* or *Fréchet-Nikodym metric* (323A (d)). Then in 521O (a) (ii), Fremlin shows that for a measure algebra whose underlying set is infinite, its density character $d(A)$ (the smallest cardinality of a dense subset) is equal to the Maharam type of $A$, $\tau(A)$. The Maharam type of $A$ is the smallest cardinality of a subset of $A$ that generates $A$ as a complete Boolean algebra (331F). 


By the Gitik-Shelah theorem (Theorem 2.6 in the reference provided by Ashutosh), for an atomless probability measure $\mu$ on $\mathcal{P}(X)$, $d(A) > \mathbf{add}(\mu)$, where $\mathbf{add}$ is the additivity. So in the setting described in the second paragraph, $\mathbf{add}(\mu) = \kappa$ by definition, so $d(A) > \kappa$. As $\kappa$ cannot be countable, this shows that $A$ cannot be separable (because separable spaces have $d(A) = \aleph\_0$, by definition). This answers the question in the title.


But we still need to deal with the question of the $\epsilon$-separated subset. Given a cardinal $\kappa$, consider the space $2^\kappa$, as the compact product space of the discrete space $2$, and equip it with the Borel $\sigma$-algebra and the unique Radon probability measure $\nu\_\kappa$ that treats $2^\kappa$ as a $\kappa$-indexed sequence of independent fair coins. For each $x \in \kappa$, define 
$$
C\_x = \{ f \in 2^\kappa \mid f(x) = 0 \}
$$
Then it should be clear that if $x \neq y$ are elements of $\kappa$, $\Delta(C\_x,C\_y) = \frac{1}{2}$, so $(C\_x)\_{x \in \kappa}$ is a $\frac{1}{2}$-separated family of Borel subsets of $2^\kappa$ of cardinality $\kappa$. This will be important later. Also, we will use the notation $\mathfrak{B}\_\kappa$ for the measure algebra of $(2^\kappa,\nu\_\kappa)$, as Fremlin does.


Let's return to the measure algebra $A$ of $(X,\mu)$. By Maharam's classification (332B), $A \cong \prod\_{i \in I}\mathfrak{B}\_{\kappa\_i}$, where $(\kappa\_i)\_{i \in I}$ is a countable family of cardinals, and under this isomorphism the measure $\mu$ is mapped to a suitable $\sigma$-convex combination of the measures $\nu\_{\kappa\_i}$, *i.e.* $\mu = \sum\_{i \in I}\alpha\_i \nu\_{\kappa\_i}$, where $\sum\_{i \in I}\alpha\_i = 1$ and $\alpha\_i > 0$ for all $i \in I$. It is easy to show that 
$$
\tau(A) \leq \prod\_{i \in I} \tau(\mathfrak{B}\_{\kappa\_i}) = \prod\_{i \in I}\kappa\_i
$$
(the fact that $\tau(\mathfrak{B}\_{\kappa\_i}) = \kappa\_i$ is 331K in Fremlin). 


We have that each $\kappa\_i \leq \tau(A)$, and since the Gitik-Shelah theorem already mentioned (543F for the specific version in Fremlin's book) shows that $\tau(A) > \kappa$, and $\kappa$ is weakly inaccessible, there must be an $i \in I$ such that $\kappa\_i > \kappa = |X|$. As $\mathfrak{B}\_{\kappa\_i}$ embeds (non-unit-preservingly) as a subalgebra of $A$, the image of $(C\_x)\_{x \in \kappa\_i}$ in $A$ is an $\epsilon = \frac{\alpha}{2}$-separated family of cardinality $\kappa\_i > |X|$ in $A$, which gives rise to such a family in $\mathcal{P}(X)$ by taking representatives of the equivalence classes.


