Packing problems are ubiquitous1, 2, ranging from oil extraction through porous rocks to grain storage in silos and the compaction of pharmaceutical powders into tablets. At a given density, particulate systems pack into a mechanically stable and amorphous jammed state3, 4. Previous theoretical studies have explored a connection between this jammed state and the glass transition4, 5, 6, 7, 8, the thermodynamics of jamming9, 10, 11, 12 and geometric modelling of random packings13, 14, 15. Nevertheless, a simple underlying mechanism for the random assembly of athermal particles, analogous to crystalline ordering, remains unknown. Here we use three-dimensional measurements of packings of polydisperse emulsion droplets to build a simple statistical model in which the complexity of the global packing is distilled into a local stochastic process. From the perspective of a single particle, the packing problem is reduced to the random formation of nearest neighbours, followed by a choice of contacts among them. The two key parameters in the model—the available space around a particle and the ratio of contacts to neighbours—are directly obtained from experiments. We demonstrate that this 'granocentric' view captures the properties of the polydisperse emulsion packing—ranging from the microscopic distributions of nearest neighbours and contacts, to local density fluctuations, to the global packing density. Application of our results to monodisperse and bidisperse systems produces quantitative agreement with previously measured trends in global density16. Our model therefore reveals a general principle of organization for random packing and may provide the foundations for a theory of jammed matter.