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In this article, we propose that the eukaryotic genome and its adaptable three\u2010dimensional packing in the form of chromatin offer a valuable template for the system architecture of DNA\u2010based digital computers. We examine embryonic and stem cells, which exhibit distinct chromatin\u2010associated condensates enriched in transcription machinery. These dynamic biomolecular condensates facilitate the spatial association of genes, genomic control elements, and molecular machinery responsible for reading the genomic code. Drawing a compelling analogy to the von Neumann computer architecture\u2014which integrates storage, processing, and memory in most electronic computers\u2014we reflect on how the operational principles of these condensates could inspire the design of a similar architecture for future DNA computers. In particular, we describe how one could recreate such an architecture by exploiting the process of surface condensation, which underlies the formation of chromatin\u2010associated condensates. We conclude by reviewing our initial steps of constructing synthetic DNA nanostructures that follow the same operational principles and enable programmable surface condensation. 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