The rational design of highly-active and stable reversible oxygen electrocatalysts for both the oxygen reduction reaction (ORR) and the oxygen evolution reaction (OER) plays a key role in rechargeable metal-air batteries, yet remains a great challenge. Herein, a novel dual-crosslinked hydrogel strategy is proposed to synthesize a new type of carbon aerogel that anchors the iron-cobalt (FeCo) particles as a bifunctional oxygen catalyst. The proposed hydrogel composed of an organic/inorganic network can be easily obtained by initiating sol-gel polymerization of cyanometalates, chitosan and graphene oxide. After pyrolysis, FeCo nanocrystals can be in situ uniformly immobilized within the N-doped "dual-network" carbon aerogels (FeCo/N-DNC) with a robust 3D porous framework. When used as an electrocatalyst, the newly developed FeCo/N-DNC aerogels exhibit a positive onset potential (0.89 V) and half-wave potential (0.81 V) for the ORR and a low overpotential (0.39 V) at 10 mA cm-2 for the OER, while presenting excellent electrochemical stability after being tested for 10 000 s. More importantly, the FeCo/N-DNC driven Zn-air battery reveals a smaller charge/discharge voltage gap, higher power/energy density and better cycling stability than the costlier Pt/C + RuO2 mixture catalyst. Our findings provide a facile and feasible synthetic strategy for obtaining highly active and stable electrocatalysts.