We demonstrate that the five-dimensional Kaluza-Klein theory of unified gravity and electromagnetism admits soliton solutions. These are regular, static and stable solutions of the field equations which correspond, upon quantization, to particles. The solitons include magnetic monopoles, which obey the Dirac quantization condition, as well as magnetic dipoles which are topologically stable. The inertial mass of the solitons is typically of order m_p/e, where m_p is the Planck mass and e the electric charge. These solitons have bizarre gravitational interactions; in fact they exert no newtonian force on slowly moving test particles, thus they have zero gravitational mass. We explain how the inequality of the gravitational and inertial masses is due to the violation of Birkhoff's theorem in Kaluza-Klein theories and is consistent with the principle of equivalence.