Most eukaryotic proteins are constituents of multiprotein complexes, yet our understanding of how individual subunits contribute to complex function, assembly and stability remains limited. Here, we utilize a doxycycline-inducible dual guide CRISPR/Cas9 system (iCas9) in mESCs to systematically investigate each component of the RNA exosome, an essential exoribonuclease complex with a central role in RNA processing and degradation. We report that depletion of individual RNA exosome components results in reciprocal proteolytic degradation of select complex subunits, revealing substantial proteostatic control mechanisms that oversees an ordered cellular assembly of the mouse RNA exosome. Phenotypic and functional analyses establish Exosc1 as a dispensable and peripheral component of the otherwise essential RNA exosome core. Our findings provide detailed insights into the cellular assembly, functions and proteostatic control of the mouse RNA exosome and offer an experimental framework to dissect other multisubunit protein complexes.