The electrocatalysis of oxygen in alkaline media is a challenging issue, influencing the performance of many electrochemical devices: fuel cells, unitized regenerative fuel cells, electrolyzers and metal-air batteries. This new manuscript proposes the synthesis of graphitic carbon nanofibers obtained by electrospinning with a cobalt-based spinel oxide, Co3O4/CNF. By means of a simple, reproducible and scalable method, a bifunctional catalyst with a promising performance is obtained, being able of carrying out the electrocatalysis of oxygen (oxidation of water, evolution and reduction of oxygen) in a basic solution. The combination of the active species on cobalt oxide (Co2+, Co3+ and Co-Nx), along with active species in the carbon nanofiber (graphitic and pyridinic N), gives rise to a catalyst with a remarkable reversibility (difference between E10 mA/cm2 (evolution) and Ehalf-wave-potential (reduction)): ΔE =795 mV), a low over-potential for the evolution of oxygen (η =416 mV) and 919 mV of oxygen reduction onset potential, very similar to that of a benchmark catalyst, Pt/C.