Closing Gmail/Outlook accounts.What is the easy way to backup all mails?

For gmail, you can try gmvault .

Use your local email program and download them?

Or use fetchmail for the same purpose and save them in a maildir or mbox to access later?

Gmail does support the normal POP and IMAP email protocols, after all. :)

just downloaded all mails from my gmail account with gmvault but "Evolution" doesen't open the .eml files.nothing happens.
gonna try fetchmail

You could use imapsync for transferring the accounts. If you have a VPS
running somewhere that would speed up transfer and you don't have to
keep your local pc running.

Usage is quite simple:
imapsync --host1 imap.truc.org --user1 foo --password1 secret1 --host2 imap.trac.org --user2 bar --password2 secret2