We have performed high-resolution angle-resolved photoemission spectroscopy on heavily electron-doped non-superconducting (SC) BaFe$_{1.7}$Co$_{0.3}$As$_2$. We find that the two hole Fermi surface pockets at the zone center observed in the hole-doped superconducting Ba$_{0.6}$K$_{0.4}$Fe$_2$As$_2$ are absent or very small in this compound, while the two electron pockets at the M point significantly expand due to electron doping by the Co substitution. Comparison of the Fermi surface between non-SC and SC samples indicates that the coexistence of hole and electron pockets connected via the antiferromagnetic wave vector is essential in realizing the mechanism of superconductivity in the iron-based superconductors.