We present a novel approach based on Bayesian field-level inference that provides representative ΛCDM initial conditions for simulation of the Local Group (LG) of galaxies and its neighbourhood, constrained by present-day observations. We extended the Bayesian Origin Reconstruction from Galaxies (BORG) algorithm with a multi-resolution approach, allowing us to reach the smaller scales needed to apply the constraints. Our data model simultaneously accounts for observations of mass tracers within the dark haloes of the Milky Way (MW) and M31, for their observed separation and relative velocity, and for the quiet surrounding Hubble flow, represented by the positions and velocities of 31 galaxies at distances between one and four megaparsec. Our approach delivers representative posterior samples of ΛCDM realisations that are statistically and simultaneously consistent with all of these observations, leading to significantly tighter mass constraints than found if the individual datasets are considered separately. In particular, we estimate the virial masses of the MW and M31 to be log₁₀(M_200c/M_⊙) = 12.07 ± 0.08 and 12.33 ± 0.10, respectively, their sum to be log₁₀(ΣM_200c/M_⊙) = 12.52 ± 0.07, and the enclosed mass within spheres of radius R to be log₁₀(M(R)/M_⊙) = 12.71 ± 0.06 and 12.96 ± 0.08 for R = 1 Mpc and 3 Mpc, respectively. The M31-MW orbit is nearly radial for most of our ΛCDM realisations, and most of them feature a dark matter sheet aligning approximately with the supergalactic plane, despite the surrounding density field not being used explicitly as a constraint. High-resolution, high-fidelity resimulations from initial conditions identified using the approximate simulations of our inference scheme continue to satisfy the observational constraints, demonstrating a route to future high-resolution, full-physics ΛCDM simulations of ensembles of LG look-alikes, all of which closely mirror the observed properties of the real system and its immediate environment.