It is shown that a strong Coulomb and exchange interactions between π electrons in a fullerene molecule lead to the localization of these electrons, quantization of their energies, and a highly correlated state manifesting itself in the formation of electron crystals. In this approximation the fullerene molecule is a nanosize quantum system consisting of a positively charged rolled-up carbon backbone surrounded by three electron crystals: (i) two crystals formed by π electrons in the 2 pz state that participate in the π bond formation through the resonance of structures and are located on the convex and concave sides of the molecule, and (ii) the crystal formed by pairs of π* electrons excited into the 2pz 3s state and participating in the formation of nonresonant π* bonds. The chemical activity and physical properties of such a system are determined by the crystal formed by electron pairs.