The supported Virtuoso database is quite esoteric ( https://en.wikipedia.org/wiki/Virtuoso_Universal_Server ).
Has one succeeded writing a script to import the data to Postgres, MySQL or Lucene?
You need a RDF/Graph database (unless you are up for a lot of re-engineering)
DBpedia is big as well.
WikiData is several magnitudes smaller than Freebase and DBpedia at the moment, but has an active healthy community.
What's the best way to run an offline copy of WikiData these days? Does it run on MySQL (or Postgres) like all other Wikimedia properties?
This isn't what happened. Freebase is still available to download and is slowly (with Google support) being migrated to WikiData[1].
There are some other pretty large Knowledge Graphs around. ConceptNet and Probase/MS Concept Graph are two that are worth looking at.
[1] https://static.googleusercontent.com/media/research.google.c...