We investigated through fully atomistic molecular dynamics simulations, the mechanical behavior (compressive and tensile) and energy absorption properties of two families (primitive (P688 and P8bal) and gyroid (G688 and G8bal)) of carbon-based schwarzites. Our results show that all schwarzites can be compressed (with almost total elastic recovery) without fracture to more than 50%, one of them can be even remarkably compressed up to 80%. One of the structures (G8bal) presents negative Poissons ratio value (auxetic behavior). The crush force efficiency, the stroke efficiency and the specific energy absorption (SEA) values show that schwarzites can be effective energy absorber materials. Although the same level of deformation without fracture observed in the compressive case is not observed for the tensile case, it is still very high (30-40%). The fracture dynamics show extensive structural reconstructions with the formation of linear atomic chains (LACs).