We investigate the bilayer Ruddlesden-Popper iridate Sr$_3$Ir$_2$O$_7$ by temperature-dependent angle-resolved photoemission. We find a narrow-gap correlated insulator, with spectral features indicative of a polaronic ground state, strikingly similar to that observed previously for the parent compounds of the cuprate superconductors. We additionally observe similar behaviour for the single-layer cousin Sr$_2$IrO$_4$, indicating that strong electron-boson coupling dominates the low-energy excitations of this exotic family of materials, and providing a microscopic link between the insulating ground states of the seemingly-disparate 3d cuprates and 5d iridates.
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