Using Outlook to save PST Files
If you are using Outlook (the actual program not OWA) you can save e-mail on your own computer using PST files. They are local backup files.
https://support.office.com/en-us/article/Introduction-to-Outlook-Data-Files-pst-and-ost-6d4197ec-1304-4b81-a17d-66d4eef30b78