If you do have a backup of your Thunderbird profile you can restore it on another computer. https://support.mozilla.org/kb/profiles-where-thunderbird-stores-user-data#w_restoring-a-profile-backup

If you were using POP email and checked he option to leave messages on the server, some messages might still be there. If you are using IMAP then the messages should still be on the server.

If the hard drive for the old computer still works you may be able to get someone to recover your Thunderbird profile from there.

Then, like christ1 states, a backup is your only hope.