We investigate the bilayer Ruddlesden-Popper iridate Sr3Ir2O7 by temperature-dependent angle-resolved photoemission. At low temperatures, we find a fully gapped correlated insulator, characterized by a small charge gap and narrow bandwidths. The low-energy spectral features show a pronounced temperature-dependent broadening and non-quasiparticle-like Gaussian line shapes. Together, these spectral features provide experimental evidence for a polaronic ground state. We observe similar behavior for the single-layer cousin Sr2IrO4, indicating that strong electron-boson coupling dominates the low-energy excitations of this exotic family of 5d compounds.