Setting up foundational OSGeo enterprise GIS?

PostGIS can do what you're talking about. I have it set up on a laptop for personal use, but if I wanted to expand it out simply dumping the data and loading it onto a more traditional server would be sufficient.

You can use Postgres's roles to define admin and read only accounts.

To back up the data you can set up a cron job that runs pg_dump and stores the results where ever you need them to go.

QGIS, OpenJump, or basically any other desktop OSGeo program can be used to access the data, produce maps, and run analysis.