In this work we demonstrate the magnetic shielding ability of a stack of YBa2Cu3O7 (YBCO) tape annuli. The annuli are cut from a 46 mm wide second generation coated conductor deposited on a Ni-5at.%W alloy ferromagnetic substrate. The inner bore of the stacked tapes is 26 mm and the outer diameter is 45 mm. Three samples with different height (24 mm, 14.9 mm, 9.9 mm) are studied. All the experiments are carried out at both room temperature and liquid nitrogen temperature (77 K). The shielding efficiency is investigated when the magnetic field is applied either parallel to the axis of the stack (axial shielding) or perpendicular to it (transverse shielding). Under an axial field, magnetic shielding is found to be effective (SF > 3) up to magnetic flux densities of 0.67 T. The presence of the ferromagnetic substrates is found to have two important consequences. First, the stack of annuli is able to shield transverse flux densities in spite of its layered structure. Second, a finite magnetic shielding effectiveness is demonstrated at room temperature. In order to understand the contribution of the ferromagnetic substrates to the shielding mechanism, we use the experimental field dependence of the magnetic permeability as determined independently from hysteresis loop measurements on the same substrates. A finite-element homogenized model solved with an H-Φ formulation is shown to successfully reproduce the shielding factor of the stack at room temperature and 77 K, both under axial and transverse applied fields. These models are also used to assess the influence of the critical current density and the magnetic permeability on the shielding efficiency. Finally, the results are used to predict the magnetic shielding properties of higher stacks, demonstrating their significant potential to shield axial fields of ∼ 0.93 T (with SF > 10) at 77 K.