It should be possible to retrieve the total number and size of all attachments on an instance.

Workaround

You can get the total count (but not size) of attachments by doing the following:

1. Navigate to "System" (by clicking the cog symbol) > "Backup Manager" > And click "Create backup for cloud"
2. Download the file when complete. Unzip the folder and open the "entities.xml" file.
3. In the "Database statistics" section you can see the Attachments count:
   

Note: Thanks to Kian for this workaround.