We employ reactive molecular-beam epitaxy to synthesize the metastable perovskite SrIrO3 and utilize in situ angle-resolved photoemission to reveal its electronic structure as an exotic narrow-band semimetal. We discover remarkably narrow bands which originate from a confluence of strong spin-orbit interactions, dimensionality, and both in- and out-of-plane IrO6 octahedral rotations. The partial occupation of numerous bands with strongly mixed orbital characters signals the breakdown of the single-band Mott picture that characterizes its insulating two-dimensional counterpart, Sr2IrO4, illustrating the power of structure-property relations for manipulating the subtle balance between spin-orbit interactions and electron-electron interactions. © 2015 American Physical Society.