The backup files are where you told Akeeba Backup to put them, i.e. in the Output directory you specified in the Configuration page.
If you haven't touched the Configuration yet, the default backup output directory is under your web site's root:
Akeeba Backup for Joomla!:
Akeeba Backup for WordPress:
Do you have Akeeba Backup for Joomla! and prefer a video instead of text? View our Downloading your backup files video tutorial. It's free and educative. Give it a try!
The easiest method is by navigating to the Manage Backups page and using the download links on the far right of each entry, but this is not the recommended method.
There have been reports that some server settings in conjunction with certain browsers may prevent you from downloading a usable archive. We recommend AGAINST using the download feature of the Manage Backups (formerly "Administer Backup Files") page for downloading your backup archive.
The suggested, tested and guaranteed method of downloading your backup files is nothing else than using FTP and its variants, such as FTPS, FTPES and SFTP. We recommend using FileZilla as your FTP client.
When you use FTP you must turn on the Binary transfer type or your backup archives will be corrupt! ALWAYS ENABLE THE BINARY TRANSFER MORE WHEN DOWNLOADING OR UPLOADING YOUR ARCHIVES THROUGH FTP, FTPS, SFTP, etc.
Once more, NEVER, EVER, USE THE "AUTO" OR "ASCII" TRANSFER MODES; ALWAYS EXPLICITLY ENABLE THE BINARY TRANSFER MODE OR YOU WILL NOT BE ABLE TO RESTORE YOUR BACKUP.
If you use FileZilla, you can ensure you are using the correct transfer mode by clicking on, and clicking the menu item before downloading your archives. If you disregard this warning and download your files through your browser, test your backups. Do not ask for support on restoring corrupt archives downloaded through the browser. We warned you for a good reason.