Back up of other email folders and contacts

Hi I work for a small print shop and was asked by my employer if I can look into a way of backing up our email, since we rely so heavily to keep track of past quotes, jobs, and job descriptions for clients. Currently we've gotten into building a large list of contacts who through old invoices we've managed to locate their info. I was wondering if there is a way to back up the contact info as well as all email files. I've tried this with my own gmail account; creating dummy draft, sent files, and contacts to see if this software would back those files up and restore them, but to no avail. I was able to get the received emails to restore. Is there something I'm not doing right or is it not possible to back up those other folders and contacts?

Thanks for your help.