Here, we introduce a 3D-DNA construction method that assembles a minimum number of DNA strands in quantitative yield, to give a scaffold with a large number of single-stranded arms. This DNA frame is used as a core structure to organize other functional materials in 3D as the shell. We use the ring-opening metathesis polymerization (ROMP) to generate block copolymers that are covalently attached to DNA strands. Site-specific hybridization of these DNA-polymer chains on the single-stranded arms of the 3D-DNA scaffold gives efficient access to DNA-block copolymer cages. These biohybrid cages possess polymer chains that are programmably positioned in three dimensions on a DNA core and display increased nuclease resistance as compared to unfunctionalized DNA cages.