To store and access a completely separate 'backed up' Thunderbird profile under Linux we can simply move the old profile directory to a safe location and add it to the Thundebrird Profile config with a suitable name.
When launching the Thunderbird backup, we should ideally force it to start in offline mode so that no information gets updated or removed when trying to communicate with the old mail server.
Identify the profile
Open Thunderbird as normal. Click on Help > Troubleshooting Information > Profiles (about:profiles).
This screen will display all information about the currently configured profiles including their system paths and name.
For this demonstration purpose we will use the fake profile name : random.default
Backup the profile
First we need to move the profile to somewhere safe and accessible (I haven't tested if renaming the profile directory causes any ill-effect).
Make sure Thunderbird is not running.
mkdir ~/.thunderbird_backup mv ~/.thunderbird/random.default ~/.thunderbird_backup/
Create a new default profile
Simply launch Thunderbird and it will create a new empty default profile for you to set up a new mail account on.
Add the backup to Thunderbird Profile
We will now go ahead and add the backed up profile to the Thunderbird Profile configuration file so that Thunderbird knows where to read the data from.
Simply add the following to the config file:
[Profile1] Name=old IsRelative=0 Path=/home/<USER>/.thunderbird_backup/random.default
Launch the backup profile
Now we can launch the backed up profile in offline mode and easily access all the previously stored mail and calendar data without worrying about losing it.
thunderbird -P old --new-instance --offline &