What is the modern equivalent to ArcInfo CLEAN?

Clean is a bunch of different things, there isn't 1 thing in the arcgis desktop / geodatabase feature class model that does all those things... but all of those bits of functionality are covered.

-if you want clean's "create polys from spagettli line work", you're looking for gp's "Data Management Tools\Features\Feature To Polygon" tool . This tool takes lines or polys and generates polygons from all closed areas.

-if you want clean's snap/cluster coincident vertices/lines together, use "Data Management Tools\Feature Class\Integrate", or you can go the create a topology and call validate on it route (it does the same thing). What this does is crack/cluster coincident lines and points.

-if you want clean's "make me feel better about the quality of my data" you run integrate AND gp's "Data Management Tools\Features\Check Geometry", which checks for polys being closed, segments being too short, polys self intersecting, etc, etc, etc.

none of the above tools require workstation. all of the above work with gdb fcs & shpfiles.

edit: reworded question: "What is the modern equivalent to a polygon coverage (and how do you clean it)?"
reworded answer: with the old cov data model you created lines (and labels) and BUILD/CLEAN'd polys from it. With the new gdb/shpfile data model you just create polygons. You don't need to BUILD/CLEAN them from other things (lines/points) you just have polys to start.

As far as "what about regions"... actually the simple feature model polygons fulfills the roles of both the poly & region data models in coverages. Cov polys were limited by strict rules (no overlap, no disconnected areas) which is why regions were created. GDB polygons are flexible, they can have many parts, overlap within a feature class, etc... then you can create a topology to add the spatial rules you desire.