Digitally encoded DNA nanostructures built via DNA self-assembly have established applications in multiplexed biosensing and storing digital information. However, a key challenge is that DNA structures are not easily copied which is of vital importance for their large-scale production and access to desired molecules by target-specific amplification. Here, we build DNA structural barcodes and demonstrate the copying and random access of the barcodes from a library of molecules using a modified polymerase chain reaction (PCR). The structural barcodes were assembled by annealing a single-stranded DNA scaffold with complementary short oligonucleotides containing protrusions as digital bits at defined locations. DNA nicks in these structures are ligated to facilitate barcode copying using PCR. To randomly access a target from a library of barcodes, we employ a non-complementary end in the DNA construct that serves as a barcode-specific primer-template. Readout of the DNA structural barcodes was performed with nanopore measurements. Our study provides a roadmap for the convenient production of large quantities of self-assembled DNA nanostructures. In addition, this strategy offers access to specific targets, a crucial capability for multiplexed single-molecule sensing and DNA data storage.