  We consider simultaneous explanations of the electron and muon $g-2$
anomalies through a $U(1)'$ extension to the Standard Model (SM) with a $Z'$.
We explore the allowed lepton$-Z'$ couplings, which are subject to various
experimental constraints, in a model-independent way. We find that only a
narrow region of parameter space with an MeV-scale $Z'$ can account for the two
anomalies. Subsequently, we explore the ability of various models to realise
these couplings, including the SM$+U(1)'$, the $N$-Higgs Doublet Model$+U(1)'$,
and a Froggatt-Nielsen style scenario. In each case, the necessary combination
of couplings cannot be obtained. It therefore seems that only rather exotic
$Z'$ models could successfully explain both $g-2$ anomalies.
