Animal tissue requires low-oxygen conditions for its maintenance. The need for low-oxygen conditions contrasts with the idea of an evolutionary leap in animal diversity as a result of expanding oxic conditions. To accommodate tissue renewal at oxic conditions, however, vertebrate animals and vascular plants demonstrate abilities to access hypoxia. Here, I argue that multicellular organisms sustain oxic conditions first after internalizing hypoxic conditions. The ‘harnessing’ of hypoxia has allowed multicellular evolution to leave niches that were stable in terms of oxygen concentrations for those where oxygen fluctuates. Since oxygen fluctuates in most settings on Earth's surface, the ancestral niche would have been a deep marine setting. The hypothesis that ‘large life’ depends on harnessing hypoxia is illustrated in the context of conditions that promote the immature cell phenotype (stemness) in animal physiology and tumour biology and offers one explanation for the general rarity of diverse multicellularity over most of Earth's history.