The Sustainable Development Goals aim to improve access to resources and services, reduce environmental degradation, eradicate poverty and reduce inequality. However, the magnitude of the environmental burden that would arise from meeting the needs of the poorest is under debate—especially when compared to much larger burdens from the rich. We show that the ‘Great Acceleration’ of human impacts was characterized by a ‘Great Inequality’ in using and damaging the environment. We then operationalize ‘just access’ to minimum energy, water, food and infrastructure. We show that achieving just access in 2018, with existing inequalities, technologies and behaviours, would have produced 2–26% additional impacts on the Earth’s natural systems of climate, water, land and nutrients—thus further crossing planetary boundaries. These hypothetical impacts, caused by about a third of humanity, equalled those caused by the wealthiest 1–4%. Technological and behavioural changes thus far, while important, did not deliver just access within a stable Earth system. Achieving these goals therefore calls for a radical redistribution of resources.