Hm, maybe I think I know where your confusion stems from.
Downloading for older versions of PHP and Joomla!, which are no longer supported by us, requires a trip to the
Compatibility page first. Find the intersection of the row with your Joomla! version and the column with the PHP version of your site. The intersection is the version number you need and is a clickable link. In your case you need
Akeeba Backup 4.0.5. Clicking on that link takes you to the download page for this old version.
Please note that this complicated process is
ONLY required if you want to use an outdated and unsupported version of our software on an outdated and unsupported version of Joomla! and / or PHP. We have
intentionally made it a tad difficult in this case: using outdated and unsupported versions of Joomla! and PHP puts your site at risk. For example, Joomla! 3.0 contains known vulnerabilities which are very actively used to hack sites. We
VERY strongly recommend upgrading your site to the latest Joomla! 3.4 version and your PHP version to the latest PHP 5.4 or 5.5 release. Anything lower can and will be used to hack your site. Moreover, by using an outdated and unsupported version of our software you will only receive limited support regarding the use of the software. We shall not provide bug fixes, new features or feature updates in old versions of our software. So here's another reason to upgrade your site as soon as possible.
Nicholas K. Dionysopoulos
Lead Developer and Director
🇬🇷Greek: native 🇬🇧English: excellent 🇫🇷French: basic • 🕐 My time zone is Europe / Athens
Please keep in mind my timezone and cultural differences when reading my replies. Thank you!