DNA origami has emerged as an exciting avenue that provides a versatile two and three-dimensional DNA-based platform for nanomedicine and drug delivery applications. Their incredible programmability, custom synthesis, efficiency, biocompatibility, and physio-chemical nature make DNA origami ideal for biomedical applications. Several recent studies demonstrated the potential of DNA origami for different technological applications, especially in drug delivery. However, several challenges related to their intracellular stability, elicitation of the immune response, and cellular fate limit the in-vivo application of these nanostructures. In this review, we critically assess the molecular-level interactions of DNA nanostructures with biological systems that will be helpful to engineer and optimize DNA nanostructures for bio applications. We highlight the hurdles that impair the potential applicability of DNA origami nanostructures in the biology and medicine field. We have also expanded the details of key strategies to overcome the limitations and extend the boundaries of DNA origami closer to nanomedicine. Finally, we explore the role Artificial Intelligence and Machine Learning techniques can play to accelerate the process of their clinical applications.