  A new layered perovskite-type oxide Ba$_2$RhO$_4$ was synthesized by a
high-pressure technique with the support of convex-hull calculations. The
crystal and electronic structure were studied by both experimental and
computational tools. Structural refinements for powder x-ray diffraction data
showed that Ba$_2$RhO$_4$ crystallizes in a K$_2$NiF$_4$-type structure,
isostructural to Sr$_2$RuO$_4$ and Ba$_2$IrO$_4$. Magnetic, resistivity, and
specific heat measurements for polycrystalline samples of Ba$_2$RhO$_4$
indicate that the system can be characterized as a correlated metal. Despite
the close similarity to its Sr$_2$RuO$_4$ counterpart in the electronic
specific heat coefficient and the Wilson ratio, Ba$_2$RhO$_4$ shows no
signature of superconductivity down to 0.16 K. Whereas the Fermi surface
topology has reminiscent pieces of Sr$_2$RuO$_4$, an electron-like
e$_g$-($d_{x^2-y^2}$) band descends below the Fermi level, making of this
compound unique also as a metallic counterpart of the spin-orbit-coupled Mott
insulator Ba$_2$IrO$_4$.
