Converting the (old) data stored inside the dBase III Plus database to any other database probably is the easy part. Converting the program and the macro's is hopeless, now even more than it was 10 years ago, because knowledge of those old tools is rapidly diminishing.
Starting all over in any modern tool to built a custom system is one option. Replacing it by a standard application package is another option, and far preferable if even remotely possible. The only prerequiste is that your business is common enough for a standard application to exist. You didn't tell anything about that.
I have a small business which has been running for years with an ancient program custom written in dBase III Plus and using a lot of Word Perfect files. When Windows first came along I was told that there was no way to convert these programs to anything which would run with Windows, especially the many macros, and that it would be very expensive to re-write all of the files. Since then things have come a long way, and I wonder whether there is any such programm today which would do this conversion. It is becoming even harder to maintain things with this old system. I presently have a Windows 7 PC as well as my old PC with DOS still working. I'm not optimistic but thought I would throw it out.