DNA nanotechnology holds substantial promise for future biomedical engineering and the development of novel therapies and diagnostic assays. The subnanometer-level addressability of DNA nanostructures allows for their precise and tailored modification with numerous chemical and biological entities, which makes them fit to serve as accurate diagnostic tools and multifunctional carriers for targeted drug delivery. The absolute control over shape, size, and function enables the fabrication of tailored and dynamic devices, such as DNA nanorobots that can execute programmed tasks and react to various external stimuli. Even though several studies have demonstrated the successful operation of various biomedical DNA nanostructures both in vitro and in vivo, major obstacles remain on the path to real-world applications of DNA-based nanomedicine. Here, we summarize the current status of the field and the main implementations of biomedical DNA nanostructures. In particular, we focus on open challenges and untackled issues and discuss possible solutions.