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We present SIBELIUS-DARK, a constrained realisation simulation of the local volume to a distance of 200~Mpc from the Milky Way. SIBELIUS-DARK is the first study of the \textit{Simulations Beyond The Local Universe} (SIBELIUS) project, which has the goal of embedding a model Local Group-like system within the correct cosmic environment. The simulation is dark-matter-only, with the galaxy population calculated using the semi-analytic model of galaxy formation, GALFORM. We demonstrate that the large-scale structure that emerges from the SIBELIUS constrained initial conditions matches well the observational data. The inferred galaxy population of SIBELIUS-DARK also match well the observational data, both statistically for the whole volume and on an object-by-object basis for the most massive clusters. For example, the $K$-band number counts across the whole sky, and when divided between the northern and southern Galactic hemispheres, are well reproduced by SIBELIUS-DARK. We find that the local volume is somewhat unusual in the wider context of $Λ$CDM: it contains an abnormally high number of supermassive clusters, as well as an overall large-scale underdensity at the level of $\approx 5$\% relative to the cosmic mean. However, whilst rare, the extent of these peculiarities does not significantly challenge the $Λ$CDM model. SIBELIUS-DARK is the most comprehensive constrained realisation simulation of the local volume to date, and with this paper we publicly release the halo and galaxy catalogues at $z=0$, which we hope will be useful to the wider astronomy community.