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A new layered perovskite-type oxide Ba$_2$RhO$_4$ was synthesized by ahigh-pressure technique with the support of convex-hull calculations. Thecrystal and electronic structure were studied by both experimental andcomputational tools. Structural refinements for powder x-ray diffraction datashowed 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, andspecific heat measurements for polycrystalline samples of Ba$_2$RhO$_4$indicate that the system can be characterized as a correlated metal. Despitethe close similarity to its Sr$_2$RuO$_4$ counterpart in the electronicspecific heat coefficient and the Wilson ratio, Ba$_2$RhO$_4$ shows nosignature of superconductivity down to 0.16 K. Whereas the Fermi surfacetopology has reminiscent pieces of Sr$_2$RuO$_4$, an electron-likee$_g$-($d_{x^2-y^2}$) band descends below the Fermi level, making of thiscompound unique also as a metallic counterpart of the spin-orbit-coupled Mottinsulator Ba$_2$IrO$_4$.

High-pressure synthesis of Ba$_2$RhO$_4$, a rhodate analogue of the layered perovskite Sr-ruthenate