Many biological tissues offer J-shaped stress-strain responses, since their microstructures exhibit a three-dimensional (3D) network construction of curvy filamentary structures that lead to a bending-to-stretching transition of the deformation mode under an external tension. The development of artificial 3D soft materials and device systems that can reproduce the nonlinear, anisotropic mechanical properties of biological tissues remains challenging. Here we report a class of soft 3D network materials that can offer defect-insensitive, nonlinear mechanical responses closely matched with those of biological tissues. This material system exploits a lattice configuration with different 3D topologies, where 3D helical microstructures that connect the lattice nodes serve as building blocks of the network. By tailoring geometries of helical microstructures or lattice topologies, a wide range of desired anisotropic J-shaped stress-strain curves can be achieved. Demonstrative applications of the developed conducting 3D network materials with bio-mimetic mechanical properties suggest potential uses in flexible bio-integrated devices.