Enhancement of specific surface area is known to improve performance of electrocatalytic system by intensifying the oxygen evolution (OER) process. In this work, nickel-cobalt oxides were formed on Bekipor ST 20AL3 3D stainless steel mesh, achieving a large surface area and high electric conductivity. Layer-by-layer potentiostatic electrodeposition was carried out in one solution by applying different potential values for individual cobalt and nickel oxide layer formation to ensure uniform distribution on each mesh filament. Hydroxide coatings were then thermally treated at 473–1073 K to obtain spinel-type oxides. X-ray diffraction analysis indicated three oxides—NiO, Co3O4 and NiCo2O4—as the main products obtained after annealing at 473–773 K. Surface morphology investigated by scanning electron microscopy showed a uniform coverage of each filament with lamellar nickel-cobalt oxides. The electrochemically active surface area of the films was determined by employing cyclic voltammetry of a ferricyanide/ferrocyanide redox system, which revealed a significant impact of annealing on the surface activity. Electrocatalytic performance tests indicated that annealing at 673 K significantly enhanced nickel-cobalt activity in OER, with overpotential of 536 mV at 10 mA cm−2, while further treatment had a detrimental effect. The coating was able to maintain this activity during long-term stability tests with no signs of surface damage. Galvanostatic charge/discharge measurements revealed a complex charge storing mechanism of nickel-cobalt oxide coatings, combining properties of batteries and supercapacitors. The highest energetic parameters achieved in this work were demonstrated by a coating annealed at 473 K (specific capacitance 769 F g−1, specific energy 312 W h kg−1, and specific power 750 W kg−1), along with great stability, maintaining 98% of its initial specific capacitance after 300 cycles.