We use a large sample of isolated dark matter halo pairs drawn from cosmological N-body simulations to identify candidate systems whose kinematics match that of the Local Group (LG) of galaxies. We find, in agreement with the ‘timing argument’ and earlier work, that the separation and approach velocity of the Milky Way (MW) and Andromeda (M31) galaxies favour a total mass for the pair of ∼5 × 1012 M⊙. A mass this large, however, is difficult to reconcile with the small relative tangential velocity of the pair, as well as with the small deceleration from the Hubble flow observed for the most distant LG members. Halo pairs that match these three criteria have average masses a factor of ∼2 times smaller than suggested by the timing argument, but with large dispersion. Guided by these results, we have selected 12 halo pairs with total mass in the range 1.6–3.6 × 1012 M⊙ for the apostle project (A Project Of Simulating The Local Environment), a suite of hydrodynamical resimulations at various numerical resolution levels (reaching up to ∼104 M⊙ per gas particle) that use the subgrid physics developed for the eagle project. These simulations reproduce, by construction, the main kinematics of the MW–M31 pair, and produce satellite populations whose overall number, luminosities, and kinematics are in good agreement with observations of the MW and M31 companions. The apostle candidate systems thus provide an excellent testbed to confront directly many of the predictions of the Λ cold dark matter cosmology with observations of our local Universe.