If you are set up as IMAP to download your email to a email client, here are some articles with advice on how to back up your email on your computer:
For Outlook and Thunderbird on Windows: http://www.pcadvisor.co.uk/how-to/software/how-back-up-your-email-3421835/
Microsofts Outlook backup support article: https://support.office.com/en-us/article/Back-up-your-email-E5845B0B-1AEB-424F-924C-AA1C33B18833
Another option with GMail is you can set it to pickup email via POP3 from the Hyper Office server. Be advised this takes time, you can’t do this last minute, and it will only get your inbox, not sent mail or subfolders. There is a description of how to do this at: https://www.lifehacker.com.au/2013/12/how-to-change-your-email-address-without-screwing-everything-up/
Mac Mail – make a copy “On My Mac”. Don’t copy 1000s of messages (i.e. select all) at a single time, do 100 at a time and give it time between copies. You can do this in all of the subfolders as well.