Using semiclassical methods, we study the bound-state spectrum in the chiral-symmetric $\ensuremath{\sigma}$ model, with $\mathrm{SU}(N)$ fermions and both with and without pions, in one space plus one time dimension. In the $\ensuremath{\sigma}$-only case, we find a rich spectrum composed of $n$ ($\ensuremath{\le}N$)-particle bound states which can be termed "normal nuclei." In a specific limit, these states approach exactly those found by Dashen, Hasslacher, and Neveu in the Gross-Neveu model, and, in general, we argue that the $\ensuremath{\sigma}$ model can be viewed as a low-energy, "effective field theory" for the Gross-Neveu model. In addition to the "normal nuclei," the $\ensuremath{\sigma}$ model contains (previously known) "shell" states---a ${\ensuremath{\varphi}}^{4}$ kink plus $n$ ($\ensuremath{\le}N$) fermions---which are analogs of the "abnormal" nuclei suggested by Lee and Wick. We discuss the relations among these "normal" and "abnormal" states; the solvability of our model allows us to examine approximation schemes used in recent field-theoretic studies of "normal" and "abnormal" nuclear matter. In the ($\ensuremath{\sigma}+\ensuremath{\pi}$) case we also find a rich spectrum of "normal nuclei." There are no kinklike states. In a specific limit the "normal nuclei" approach exactly those found by Shei in the chiral Gross-Neveu model. However, since our semiclassical results predict in this case the spontaneous breakdown of a continuous symmetry in two dimensions, their interpretation remains unclear. We speculate on the implications of our predictions for theories with small explicit symmetry breaking and on the mechanism by which the semiclassical results could be altered by quantum-fluctuations corrections.