Structural and electronic properties of oligothiophene nanowires and rings synthesized on a Au(111) surface are investigated by scanning tunneling microscopy. The spectroscopic data of the linear and cyclic oligomers show remarkable differences which, to a first approximation, can be accounted by considering electronic state confinement to one-dimensional boxes having, respectively, fixed and periodic boundary conditions. A more detailed analysis shows that polythiophene must be treated as a ribbon (i.e., having an effective width) rather than a purely 1D structure. A fascinating consequence is that the molecular nanorings act as whispering gallery mode resonators for electrons, opening the way for new applications in quantum electronics.