There are few known semiconductors exhibiting both strong optical response and large dielectric polarizability. Inorganic materials with large dielectric polarizability tend to be wide-band gap complex oxides. Semiconductors with a strong photoresponse to visible and infrared light tend to be weakly polarizable. Interesting exceptions to these trends are halide perovskites and phase-change chalcogenides. Here we introduce complex chalcogenides in the Ba-Zr-S system in perovskite and Ruddlesden-Popper structures as a family of highly polarizable semiconductors. We report the results of impedance spectroscopy on single crystals that establish BaZrS₃ and Ba₃Zr₂S₇ as semiconductors with a low-frequency relative dielectric constant ɛ0 in the range 50–100 and band gap in the range 1.3–1.8 eV. Our electronic structure calculations indicate that the enhanced dielectric response in perovskite BaZrS₃ versus Ruddlesden-Popper Ba₃Zr₂S₇ is primarily due to enhanced IR mode-effective charges and variations in phonon frequencies along 〈001〉; differences in the Born effective charges and the lattice stiffness are of secondary importance. This combination of covalent bonding in crystal structures more common to complex oxides, but comprising sulfur, results in a sizable Fröhlich coupling constant, which suggests that charge carriers are large polarons