Is it possible to select multiple email messages, and then download the files in their attachments in one archive file?
The native way to click to open the emails one by one, and click to download their attachments individually is tedious.
It does not really answer your question for existing mail, but going forward it is really convenient to setup a "receipe" on IFTTT. You can create a rule (e.g. based on sender) and have attachments saved to Google Drive automatically.
You might try "Mail Attachment Downloader" http://gearmage.com/maildownloader.html
It doesn't work on selected mails, but you can filter by subject, sender, etc.
There are options to select certain labels, but I couldn't get it working.
It did the job for getting emailed snapshots from a security camera into one folder.
Gmail has an extension that can download them to your Google Drive.
I haven't used it but it has quite a few users.
I was also questioning same to myself and then I developed https://github.com/munir131/attachment-downloader small utility to solve this problem.
It allow me to download all attachment in bulk. If you have little knowledge about how to run node application then it will be solve your problem.