Within the one-dimensional Hubbard model linear closed chains with various numbers of sites are considered in the self-consistent random phase approximation ͑SCRPA͒. Excellent results with a minimal numerical effort are obtained for ͑2+4n͒-site cases, confirming earlier results with this theory for other models. However, the 4n-site cases need further consideration. The SCRPA solves the two-site problem exactly. It therefore contains the two-electron and high-density Fermi gas limits correctly.