Long-ranged van der Waals (vdW) interactions are most often treated via Lennard-Jones approaches based on the combination of two-body and dipolar approximations. While beyond-dipole interactions and many-body contributions were separately addressed, little is known about their combined effect, especially in large molecules and relevant nanoscale systems. Here, we provide a full many-body description of vdW interactions beyond the dipole approximation, efficiently applicable to large-scale systems. Dipole-quadrupole interactions consistently exhibit large magnitude up to nm-scale separations, while many-body effects lead to system-dependent screening effects, which can reduce vdW interactions by a large fraction. Combined many-body and multipolar terms emerge as an essential ingredient for the reliable description of vdW interactions in molecular and nanoscale systems.