Three-dimensional cell and organoid cultures rely on the mechanical support of viscoelastic matrices. However, commonly used matrix materials lack control over key cell-instructive properties. Here we report on fully synthetic hydrogels based on DNA libraries that self-assemble with ultrahigh-molecular-weight polymers, forming a dynamic DNA-crosslinked matrix (DyNAtrix). DyNAtrix enables computationally predictable and systematic control over its viscoelasticity, thermodynamic and kinetic parameters by changing DNA sequence information. Adjustable heat activation allows homogeneous embedding of mammalian cells. Intriguingly, stress-relaxation times can be tuned over four orders of magnitude, recapitulating mechanical characteristics of living tissues. DyNAtrix is self-healing, printable, exhibits high stability, cyto- and haemocompatibility, and controllable degradation. DyNAtrix-based cultures of human mesenchymal stromal cells, pluripotent stem cells, canine kidney cysts and human trophoblast organoids show high viability, proliferation and morphogenesis. DyNAtrix thus represents a programmable and versatile precision matrix for advanced approaches to biomechanics, biophysics and tissue engineering.