Comment by Gro-Tsen on Reference: If $X$ is metrizable, then $X$ is realcompact iff $|X|$ is non-measurable

I'm not sure whether the difficult part is “if” or “only if”, but just in case, the “if” part (i.e., metrizable of non-measurable cardinality implies realcompact) is in Gillman & Jerison, Rings of Continuous Functions (1960), theorem 15.24 on page 232.