The paper considers models of porous structures based on triply periodic minimal surfaces, which are an example of additively produced metamaterials of a new topological class. A numerical technology for the construction of metamaterial periodicity cell based on surface structures is developed, which ensures the periodic boundary conditions fulfilment on finite element meshes. The elastic properties of metamaterials have been calculated by the method of direct numerical homogenization at the periodicity cell meso-level. The dependences of the effective properties on the volume fraction of the metamaterial solid phase are revealed and it is noted that their proper description requires an orthotropic material model. It is shown that the considered types of metamaterials demonstrate strongly nonlinear dependence of elastic properties on the relative density or volume fraction of the solid phase of the metamaterial. The curvature of the curve is more pronounced for values of relative density less than 50 %, which may indicate a pronounced influence of the topological characteristics of the cell on the behaviour of the metamaterial at the meso-level. When analysing the Poisson’s ratios, a significant variation in their behaviour for different types of metamaterials is observed. The reason for this phenomenon may be the more pronounced influence of the unit cell topology on the transverse deformations. The consequence of this phenomenon is the apparent existence of stationary points corresponding to the maximum or minimum achievable values of the Poisson’s ratio, which can be useful in problems where its value has a significant influence on the global result.