  We investigate level sets of the Gaussian free field on continuous transient
metric graphs $\tilde{\mathcal G}$ and study the capacity of its level set
clusters. We prove, without any further assumption on the base graph
$\mathcal{G}$, that the capacity of sign clusters on $\tilde{\mathcal G}$ is
finite almost surely. This leads to a new and effective criterion to determine
whether the sign clusters of the free field on $\tilde{\mathcal G}$ are bounded
or not. It also elucidates why the critical parameter for percolation of level
sets on $\tilde{\mathcal G}$ vanishes in most instances in the massless case
and establishes the continuity of this phase transition in a wide range of
cases, including all vertex-transitive graphs. When the sign clusters on
$\tilde{\mathcal G}$ do not percolate, we further determine by means of
isomorphism theory the exact law of the capacity of compact clusters at any
height. Specifically, we derive this law from an extension of Sznitman's
refinement of Lupu's recent isomorphism theorem relating the free field and
random interlacements, proved along the way, and which holds under the sole
assumption that sign clusters on $\tilde{\mathcal G}$ are bounded. Finally, we
show that the law of the cluster capacity functionals obtained in this way
actually characterizes the isomorphism theorem, i.e. the two are equivalent.
