Deposition of Co atoms on Au(111) surfaces leads to the formation of self-organized bilayer Co nanowires at the step edges between adjacent narrow Au(111) terraces. Scanning tunneling microscopy and spectroscopy at low temperatures is used to probe the influence of the finite dimensions on the local density of states for both the Co wires and the narrow Au terraces. Confinement of Au surface state electrons to narrow Au terraces causes a shift of the Au surface state onset energy to higher energies. For the Co nanowires discrete energy levels are observed. Moreover, standing wave patterns occur at the surface of both the Au and the Co. The patterns increase in complexity with increasing energy. All Co nanowires formed at the edges of narrow Au terraces reveal a characteristic maximum in the local density of states at a significantly different energy when compared to the Co islands that are formed on large Au terraces. The experimental results can be interpreted in terms of a simple particle-in-a-box model.