Studies of nanoscale superconducting structures have revealed various physical phenomena and led to the development of a wide range of applications. Most of these studies concentrated on one- and two-dimensional structures due to the lack of approaches for creation of fully engineered three-dimensional (3D) nanostructures. Here, we present a 'bottom-up' method to create 3D superconducting nanostructures with prescribed multiscale organization using DNA-based self-assembly methods. We assemble 3D DNA superlattices from octahedral DNA frames with incorporated nanoparticles, through connecting frames at their vertices, which result in cubic superlattices with a 48 nm unit cell. The superconductive superlattice is formed by converting a DNA superlattice first into highly-structured 3D silica scaffold, to turn it from a soft and liquid-environment dependent macromolecular construction into a solid structure, following by its coating with superconducting niobium (Nb). Through low-temperature electrical characterization we demonstrate that this process creates 3D arrays of Josephson junctions. This approach may be utilized in development of a variety of applications such as 3D Superconducting Quantum interference Devices (SQUIDs) for measurement of the magnetic field vector, highly sensitive Superconducting Quantum Interference Filters (SQIFs), and parametric amplifiers for quantum information systems.