Frustrated magnetic materials, in which local conditions for energy minimization are incompatible because of the lattice structure, can remain disordered to the lowest temperatures. Such is the case for Ba₃CuSb₂O₉, which is magnetically anisotropic at the atomic scale but curiously isotropie on mesoscopic length and time scales. We find that the frustration of Wannier's Ising model on the triangular lattice is imprinted in a nanostructured honeycomb lattice of Cu⁺ ions that resists a coherent static Jahn-Teller distortion. The resulting two-dimensional random-bond spin-1/2 system on the honeycomb lattice has a broad spectrum of spin-dimer-like excitations and low-energy spin degrees of freedom that retain overall hexagonal symmetry.