The quantum dynamics of low-energy electron scattering from C-60 molecules is carried out using the full anisotropic interaction between the molecule at its equilibrium geometry and the impinging electron. The interaction is constructed using a nonempirical model potential for the short-range dynamical correlation between the scattered electron and the bound electrons, while both the static interaction and the nonlocal exchange interaction are treated correctly. The fairly large size of the carbon cage, and the hollow structure which is created inside it, are unique features of the C-60 molecule that are shown here to play a very important role in forming a broad variety of metastable electron-attachment states without electronic excitation being involved. In spite of various aspects of the resonant process which are not considered in the present treatment, the calculations show very clearly the occurrence of several trapping processes for the scattered electrons which could be amenable to experimental observation.