One of the chief paradoxes of molecular oxygen (O2) is that it is an essential requirement for multicellular eukaryotes on Earth while simultaneously posing a threat to their survival via the formation of reactive oxygen species. In this paper, the constraints imposed by O2 on Earth-like complex life are invoked to explore whether worlds with abiotic O2 inventories can harbor such organisms. By taking the major O2 sources and sinks of Earth-like planets into account using a simple model, it is suggested that worlds that receive time-averaged X-ray and extreme ultraviolet fluxes that are ≳10 times higher than Earth might not be capable of hosting complex lifeforms because the photolysis of molecules such as water may lead to significant O2 buildup. Methods for testing this hypothesis by searching for anticorrelations between biosignatures and indicators of abiotic O2 atmospheres are described. In the event, however, that life successfully adapts to high-oxygen environments, these worlds could permit the evolution of large and complex organisms.