We employ reactive molecular-beam epitaxy to synthesize the metastable
perovskite SrIrO3 and utilize {\it 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