The halo of the Milky Way provides a laboratory to study the properties of the shocked hot gas that is predicted by models of galaxy formation. There is observational evidence of energy injection into the halo from past activity in the nucleus of the Milky Way(1-4); however, the origin of this energy (star formation or supermassive-black-hole activity) is uncertain, and the causal connection between nuclear structures and large-scale features has not been established unequivocally. Here we report soft-X-ray-emitting bubbles that extend approximately 14 kiloparsecs above and below the Galactic centre and include a structure in the southern sky analogous to the North Polar Spur. The sharp boundaries of these bubbles trace collisionless and non-radiative shocks, and corroborate the idea that the bubbles are not a remnant of a local supernova(5) but part of a vast Galaxy-scale structure closely related to features seen in gamma-rays(6). Large energy injections from the Galactic centre(7) are the most likely cause of both the gamma-ray and X-ray bubbles. The latter have an estimated energy of around 1056 erg, which is sufficient to perturb the structure, energy content and chemical enrichment of the circumgalactic medium of the Milky Way.