Making local measurements on an entangled quantum many-body state is expected to reduce quantum entanglement. However, recently it was found that in the thermodynamic limit it is possible to robustly protect quantum entanglement from continuous local measurements. I will discuss a simplified model of a repeatedly measured interacting many-body quantum system, and show how it preserves entanglement up to time exponential in the number of degrees of freedom, after which the state purifies. I will also show that in a model of free fermions, purification occurs in polynomial time, consistent with the idea that interactions are crucial for preserving the entanglement.