Architected materials achieve unique mechanical properties through precisely engineered microstructures that minimize material usage. However, a key challenge of low-density materials is balancing high stiffness with stable deformability up to large strains. Current microstructures, which employ slender elements such as thin beams and plates arranged in periodic patterns to optimize stiffness, are largely prone to instabilities, including buckling and brittle collapse at low strains. This challenge is here addressed by introducing a new class of aperiodic architected materials inspired by quasicrystalline lattices. Beam networks derived from canonical quasicrystalline patterns, such as the Penrose tiling in two dimensions and icosahedral quasicrystals (IQCs) in three dimensions, are shown to create stiff, stretching-dominated topologies with non-uniform force chain distributions, effectively mitigating the global instabilities observed in periodic designs through distributed localized buckling instabilities. Numerical and experimental results confirm the effectiveness of these designs in combining stiffness and stable deformability at large strains, representing a significant advancement in the development of low-density metamaterials for applications requiring high impact resistance and energy absorption. These results demonstrate the potential of deterministic quasi-periodic topologies to bridge the gap between periodic and random structures, while branching toward uncharted territory in the property space of architected materials.