Reaching a technological advantage with large quantum systems requires safeguarding their many-body entanglement. Dissipation typically acts to decohere a quantum system via random projective noise but, when judiciously engineered together with coherent interactions, it can funnel the system towards a target entangled state. The native interactions and dissipative channels available to most systems are, however, difficult to combine effectively. Here we propose interleaving Floquet Hamiltonian engineering, which allows the construction of non-native coherent interactions, with weak measurement, which enables a tuneable dissipative channel, to enable programmable and autonomous stabilization of many-body entanglement. We show this analytically and numerically for the central-spin system of a semiconductor quantum dot, for which we construct spin-squeezed and Schrödinger-cat states that are stabilized against realistic levels of dephasing. Our approach is applicable to any system compatible with periodic drives and tuneable measurement strength and could enable novel approaches to practical error correction.