you just need to use the analysis tools, extract, clip function.
The input feature is your geodatabase, the clip feature is the rectangular feature
I know this is an old thread, but I thought I would update. If you have access to the production mapping extension, you can use the "Production Clip" tool to clip an entire database.
Another option is to use Distributed Geodatabase and use the Extract Data - tool https://desktop.arcgis.com/en/arcmap/latest/manage-data/geodatabases/copy-geodatabase-schema-with-ex... - Pick Data, and Show Advanced Options for overriding data extraction defaults when I click Next. You then have an option to clip the all/any database content into a new .gdb, clipped to current extent, a specified extent or boundary of selected graphic.