You can have offline copy of all your emails with attachments using just Thunderbird.

1. Connect to your webmail account using IMAP (so that the server retains a copy of your mail).

2. Enable Allow remote content in messages under Privacy section of the Preference if you want images in the email to be downloaded as well. (See note below).

3. Make sure that the you have the following settings under Synchronization & Storage of your Account Settings.
   3.1 Enable Keep messages for this account on this computer.
   3.2 Under Advanced enable all the folders that needs to be synchronized.
   3.3 Under Disk Space choose Synchronize all messages locally regardless of age.
   3.4 Enable Don't delete any messages.

4. Now click the computer icon on the lower bottom corner of Thunderbird or choose File > Offline > Work Offline. A popup will be displayed. Choose Download Now.

Thunderbird will now download all your emails and attachments offline.

NB: The only caveat is that Thunderbird won't download remote contents like images from unknown senders even if you enable remote content (step 2). The sender has to be present in your address book or you need to whitelist the domains. See instructions here. This is for privacy reasons.

Privacy

1. It's better to have full disk encryption (LUKS) and home directory encryption enabled or at least if mails need to be stored securely.

2. Enable master password in Thunderbird. Install StartupMaster extension. It fixes the multiple password prompt if you have multiple email accounts added.