  We report on the synthesis and the superconductivity of Zr$_4$Rh$_2$O$_{x}$
($x$ = 0.4, 0.5, 0.6, 0.7, 1.0). These compounds crystallize in the
$\eta$-carbide structure, which is a filled version of the complex
intermetallic Ti$_2$Ni structure. We find that in the system
Zr$_4$Rh$_2$O$_{x}$, already a small amount ($x$ $\geq$ 0.4) of oxygen addition
stabilizes the $\eta$-carbide structure over the more common intermetallic
CuAl$_2$ structure-type, in which Zr$_2$Rh crystallizes. We show that
Zr$_4$Rh$_2$O$_{0.7}$ and Zr$_4$Rh$_2$O are bulk superconductors with critical
temperatures of $T_c \approx$ 2.8 K and 4.7 K in the resistivity, respectively.
Our analysis of the superconducting properties reveal both compounds to be
strongly type-II superconductors with critical fields up to $\mu_0 H_{c1}$(0)
$\approx$ 8.8 mT and $\mu_0 H_{c2}$(0) $\approx$ 6.08 T. Our results support
that the $\eta$-carbides are a versatile family of compounds for the
investigation of the interplay of interstitial doping on physical properties,
especially for superconductivity.
