Context. Understanding circumstellar disks is of prime importance in astrophysics; however, their birth process remains poorly constrained due to observational and numerical challenges. Recent numerical works have shown that the small-scale physics, often wrapped into a sub-grid model, play a crucial role in disk formation and evolution. This calls for a combined approach in which both the protostar and circumstellar disk are studied in concert.Aims. We aim to elucidate the small-scale physics and constrain sub-grid parameters commonly chosen in the literature by resolving the star-disk interaction.Methods. We carried out a set of very high resolution 3D radiative-hydrodynamics simulations that self-consistently describe the collapse of a turbulent, dense molecular cloud core to stellar densities. We studied the birth of the protostar, the circumstellar disk, and its early evolution (< 6 yr after protostellar formation).Results. Following the second gravitational collapse, the nascent protostar quickly reaches breakup velocity and sheds its surface material, thus forming a hot (∼103 K), dense, and highly flared circumstellar disk. The protostar is embedded within the disk such that material can flow without crossing any shock fronts. The circumstellar disk mass quickly exceeds that of the protostar, and its kinematics are dominated by self-gravity. Accretion onto the disk is highly anisotropic, and accretion onto the protostar mainly occurs through material that slides on the disk surface. The polar mass flux is negligible in comparison. The radiative behavior also displays a strong anisotropy, as the polar accretion shock was shown to be supercritical, whereas its equatorial counterpart is subcritical. We also find a remarkable convergence of our results with respect to initial conditions.Conclusions. These results reveal the structure and kinematics in the smallest spatial scales relevant to protostellar and circumstellar disk evolution. They can be used to describe accretion onto regions commonly described by sub-grid models in simulations studying larger-scale physics.