The electron-positron cluster, a new physical object is investigated. In the metallic phase clusters considered consist of equal numbers of delocalized electrons and positrons bounded by attractive Coulomb potential and the potential of particles mutual polarization. We focus on investigating the cluster properties dependent on their size (up to one hundred electron-positron pairs). Our calculations have been carried out within the Hartree - Fock approximation in which clusters are bounded by nonlocal exchange interactions. The set of the basis wave functions has been obtained by means of the self-consistent procedure taking into account the interactions between cluster particles of both similar and opposite charges.