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 the later to keep your mails safer.
2. Enable master password in Thunderbird. Install StartupMaster extension. It fixes the multiple password prompt if you have multiple email accounts added.