We have developed a semi-empirical simulation of the extragalactic radio continuum sky suitable for aiding the design of next generation radio interferometers such as the Square Kilometre Array (SKA). The emphasis is on modelling the large-scale cosmological distribution of radio sources rather than the internal structure of individual galaxies. Here we provide a description of the simulation to accompany the online release of a catalogue of similar or equal to 320 million simulated radio sources. The simulation covers a sky area of 20x20 deg^2 - a plausible upper limit to the instantaneous field of view attainable with future (e.g. SKA) aperture array technologies - out to a cosmological redshift of z=20, and down to flux density limits of 10 nJy at 151, 610 MHz, 1.4, 4.86 and 18 GHz. Five distinct source types are included: radio-quiet active galactic nuclei (AGN), radio-loud AGN of the Fanaroff-Riley type I (FR I) and FR II structural classes, and star-forming galaxies, the latter split into Populations of quiescent and starbursting galaxies.