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 $\mathrm{BaZr}{S}_3$ and ${\mathrm{Ba}}_3{\mathrm{Zr}}_2{S}_7$ as semiconductors with a low-frequency relative dielectric constant ${\varepsilon }_0$ 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 $\mathrm{BaZr}{S}_3$ versus Ruddlesden-Popper ${\mathrm{Ba}}_3{\mathrm{Zr}}_2{S}_7$ 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.