This study focuses on the design, additive manufacturing, and characterization of porous carbon-based structures in the form of a triply periodic minimal surface (TPMS) with outstanding mechanical properties and oxidation resistance, combined with good electrical and thermal conductivity. Binder jetting (BJ) of graphite-carbon black powders was used to 3D print computational models of three TPMS with different topologies and geometric porosities. Infiltration and pyrolysis (PIP) with furan resin was then performed to densify the parts. Composite materials, comprising a highly disordered carbon matrix binding well-crystallized graphite grains, were obtained. The printed and pyrolyzed samples are highly porous TPMS cylinders with diameter, height and a surface thickness of ~19 mm, ~33 mm and 0.76 mm, respectively. The samples have a skeleton intrinsic porosity of 20%, of which 8% is open, meaning that the material could potentially be further infiltrated and densified. Nevertheless, the samples have better mechanical properties than compressed carbon-graphite composites, as well as 3D-printed carbon produced by direct ink writing and stereolithography. High-temperature tests showed that, although the amorphous carbon matrix is more prone to oxidation than the graphite grains, the overall oxidation resistance remains exceptionally high. These properties allow for applications as Joule resistors and in seasonal thermal storage.