If you go to menu Project->Statistics… then ShipWeight will display a window listing all projects and you can also see how many items (rows) each project has. See the picture below. Since a database and projects share the same file(s) on the server, there is no data size related to a specific project, but the number of items will of course be a good indication of the space it is using.
You can print a report from this window, or you can rightclick and copy the information and paste to excel.
Note that it may take a few minutes for this window to pop up depending on the number of projects, etc.