I worked at a company once with huge monolith system, mainly revolving around a relational database. We tried for years to break out of this, and if my understanding is correct this organization is still on this monolith today. There were a number of challenges, and we tried a number of different solutions (NoSQL models/object stores/etc), but it felt like there were base level assumptions about the availability of data in the core application that felt impossible to address without a full scale rewrite and reevaluation of all previous assumptions.
Perhaps I've answered my own question here -- it just needed to be completely redesigned from the ground up. Short of doing that however, would anyone care to provide high level insight on how they'd break down this problem, and what technology they might use to address it?