We report a detailed experimental and theoretical study on the effect of hydrostatic pressure on various structural and magnetic aspects of the layered honeycomb antiferromagent α-RuCl3. Through measurements of the magnetic susceptibility χ performed under almost ideal hydrostatic-pressure conditions by using helium as a pressure-transmitting medium, we find that the phase transition to zigzag-type antiferromagnetic order at TN = 7.3 K can be rapidly suppressed to about 6.1 K at a weak pressure of about 94 MPa. A further suppression of TN with increasing pressure is impeded, however, due to the occurrence of a pressure-induced structural transition at p≥ 104 MPa, accompanied by a strong dimerization of Ru-Ru bonds, which gives rise to a collapse of the magnetic susceptibility. Whereas the dimerization transition is strongly first order, the magnetic transition under varying pressure and magnetic field also reveals indications for a weakly first-order transition. We assign this observation to a strong magnetoelastic coupling in this system. Measurements of χ under varying pressure in the paramagnetic regime (T>TN) and before dimerization (p< 100 MPa) reveal a considerable increase of χ with pressure. These experimental observations are consistent with the results of ab initio density functional theory (DFT) calculations on the pressure-dependent structure of α-RuCl3 and the corresponding pressure-dependent magnetic model. We find that pressure strengthens the nearest-neighbor Heisenberg J and off-diagonal anisotropic Γ coupling and simultaneously weakens the Kitaev K and anisotropic Γ′ coupling. Comparative susceptibility measurements on a second crystal showing two consecutive magnetic transitions instead of one, indicating the influence of stacking faults, reveal that by the application of different temperature-pressure protocols the effect of these stacking faults can be temporarily overcome.