We investigated the stellar mass-metallicity relation (MZR) using a sample of 637 quiescent galaxies with 10.4 ≤ log(M_*/M_⊙) < 11.7 selected from the LEGA-C survey at 0.6 ≤ z ≤ 1. We derived mass-weighted stellar metallicities using full-spectral fitting. We find that while lower-mass galaxies are both metal-rich and metal-poor, there are no metal-poor galaxies at high masses, and that metallicity is bounded at low values by a mass-dependent lower limit. This lower limit increases with mass, empirically defining a MEtallicity-Mass Exclusion (MEME) zone. We find that the spectral index MgFe ≡ √Mgb × Fe4383, a proxy for the stellar metallicity, also shows a mass-dependent lower limit resembling the MEME relation. Crucially, MgFe is independent of stellar population models and fitting methods. By constructing the metallicity enrichment histories, we find that, after the first gigayear, the star formation history of galaxies has a mild impact on the observed metallicity distribution. Finally, from the average formation times, we find that galaxies populate differently the metallicity-mass plane at different cosmic times, and that the MEME limit is recovered by galaxies that formed at z ≥ 3. Our work suggests that the stellar metallicity of quiescent galaxies is bounded by a lower limit which increases with the stellar mass. On the other hand, low-mass galaxies can have metallicities as high as galaxies ∼1 dex more massive. This suggests that, at log(M_*/M_⊙)≥10.4, rather than lower-mass galaxies being systematically less metallic, the observed MZR might be a consequence of the lack of massive metal-poor galaxies.