Is it possible to come up with a measure space $(X, \mathcal{M}, \mu)$ such that $$\{\mu(E) \mid E \in \mathcal{M}\} = [0,1] \cup [3,4]$$
I suspect yes, although I have no idea how to construct such a measure space, any attempt to create this "break" in the image of measures seems to be insurmountable due to countable additivity.
Consider e.g. $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ with the measure$$\mu(E) := \lambda(E \cap [0,1]) + 3 \delta_{2}(E),$$ where $\lambda$ denotes the Lebesgue measure and $\delta_x$ is the Dirac measure at $x$.
Remark: If $\mu$ is a finite measure without atoms, then the range $\{\mu(E); E \in \mathcal{M}\}$ is convex. You are looking for a measure with range $[0,1] \cup [3,4]$. Since this is clearly not a convex set, this means that the measure needs to have an atom. The Dirac measure is the simplest way to create an atom.