To answer your question: I’d use Libre Office with Firebird (as an embedded engine) for this. If you’d already got that far, and meant more subtle things by “… what type…” of database, then see some of the comments already made by others. But I would START with a version of the system that ran on just one computer, using a simple, one user access, local installation of Libre Office. The next stage would be to have several such workstations, and use pedestrian techniques to update the separate copies of the database whenever necessary. A system built thus would be a perfectly sensible first strp along a road that could one day lead to one file on a central server which was the source of the data for any PC on the LAN.
Firebird is currently “hidden”… see other. I believe this is NOT because it is, in general, problematic, but because the tools to convert (dependably) an old HSQL database (ANY old one) to a Firebird database are not yet available. I doubt they ever will be. A bridge too far. But I also believe that HSQL’s days are numbered, that Firebird is the future. I speak as someone who has had to redo databases to migrate to the “latest, greatest” many times. Leaving Borland’s excellent Paradox was not my first such migration. See the Old Fart’s Guide to the History of Computers. They didn’t use Paradox at Bletchley, but it came not long after that.
To add things that I think are relevant: Big visions are wonderful things… but businesses thrive on things that work. And no business should let itself become too dependant on one cog in the machine. The following isn’t a big vision, but it might serve as a starting point for your ambitions. A very do-able starting point.
Set up a small database that has at its heart a field with multi-line text. The other fields would provide ways to search for the record you need. Always keep an up-to-date ink-on-paper copy of the data available for when the computers aren’t working, and as an ultimate backup against the day the disk crashes. But, day to day, a user needs to enter name and address on an invoice? The form from the database runs in a small window at the edge of your screen, one field lets the user enter search criteria, another displays matching records’ multiline text fields. A quick select/copy/paste between windows and the job is done. Get that much working, and it would probably be easy enough to add a button to say “Copy the text in the ‘contact’ field.”