Strain applied to a condensed-matter system can be used to engineer its excitation spectrum via artificial gauge fields or it may tune the system through transitions between different phases. Here we demonstrate that strain tuning of the ground state of otherwise highly degenerate frustrated systems can induce novel phases, both ordered and disordered. For the classical Heisenberg antiferromagnet on the kagome lattice, we show that weak triaxial strain reduces the degeneracies of the system, leading to a classical spin liquid with noncoplanar configurations, while stronger strain drives the system into a highly unconventional state which displays signatures of both spin-glass behavior and magnetic long-range order. We provide experimentally testable predictions for the magnetic structure factor, characterize the ground-state degeneracies and the excitation spectrum, and analyze the influence of sample shape and boundaries. Our paper opens the way to strain engineering of highly frustrated magnets.