The closed-shell Generator Coordinate Dirac-Fock method is applied to perform Dirac-Fock-Coulomb and Dirac-Fock-Breit calculations for the He atom and the helium-like ionic species Ne⁺⁸^{+8}, Ar⁺¹⁶^{+16}, and Sn⁺⁴⁸^{+48}. With the Generator Coordinate Dirac-Fock method, the integral Dirac-Fock equations are integrated numerically so as to generate Gaussian basis sets for the atomic species under study. The Dirac-Fock-Coulomb and Dirac-Fock-Breit energies obtained here for He and He-like ionic species with the Generator Coordinate Dirac-Fock formalism are better than the corresponding energies obtained with variational calculations using even-tempered Gaussian-type functions. The Dirac-Fock-Coulomb energies obtained for Ne⁺⁸^{+8} and Ar⁺¹⁶^{+16} with the Generator Coordinate Dirac-Fock method are in excellent agreement with Desclaux's numerical calculations, and for He and Sn⁺⁴⁸^{+48} our Dirac-Fock-coulomb energy results are lower than the corresponding energies obtained with Desclaux's numerical-finite-difference program.