The original question makes no sense to me, in that the options don't belong together. BerkeleyDB is a database engine only, while Access is an application development tool that ships with a default file-based (i.e., non-server) database engine (Jet). By virtue of putting Access with Berkeley, it seems obvious that what is needed is only a database engine, and no application at all, but how end users use Berkeley DB without a front end, I don't know (I've only used it from the command line).
Those who cannot run a Jet MDB with 20 simultaneous users are simply not competent to be giving advice on using Jet as a data store. It is completely doable as long as best practices are followed. I would recommend in addition to Microsoft's Best Practices web page, Tony Toews's Best Practices, and Tony's Corruption FAQ (i.e., things you want to avoid doing in order to have a stable application).
I strongly doubt that the original questioner is building no front end application, but since he doesn't indicate what kind of front end is involved, it's hard to recommend a back end that will go with it. Access has the advantage of giving you both parts of the equation, and when used properly, is perfectly reliable for multiple users.
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David W. Fenton
http://dfenton.com/DFA/