The work of Skyrme has shown that the SU(2)×SU(2) chiral model has nontrivial topological sectors which admit solitons for generic chiral Lagrangians. In this paper, we study such models in the presence of baryon fields. The baryon number and strangeness of the solitons, and the bound states of the nucleon to the soliton are investigated. It is found that long-lived levels with large baryon number B and strangeness (≳6 in magnitude) and masses somewhere in the range 1.8 to 5.6 GeV must exist. Some of these levels have half-integral electric charge and exotic relation between B and spin s (e.g., even B and half-integer s). It is speculated that these levels may be related to the anomalous nuclei whose existence has been confirmed in cosmic-ray and LBL Bevalac experiments.