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 I did a full backup about an hour ago, shortly after updating to Akeeba 5.0.4. It worked without any issues.
I was doing some template and design testing, and now simply want to go back to that last backup.
In the "Manage Backups" window I select that most recent backup checkbox and click the Restore button.
The next page page asks only about extraction method (I choose "Write directly to files") and about encryption, which I didn't use.
When I click Start Restoration, it goes the page
"Archive extraction in progress" - showing Bytes read, Bytes extracted, and Files extracted and sits there indefinitely. (see attached file).
When I check on the hosting account, it has over 60GB of free space, and there is no appreciable server load. There is no sign that anything is happening at all.
The longest I've waited so far is 30 minutes, with no sign that anything is happening.

Thanks for your reply.
So I decided to attempt the restore to a new folder, just like when I move joomla installs to a new site, and then upgrade akeeba to the beta version you recommend (or wait for a regular update).

I created a new folder, new db, etc., copied the 5.0.4 backup to the new folder, ran kickstart.php, removed the two offending files (installation/platform/steps.php and installation/platform/iniprocess.php) and then ran the installer.
It works through to the Site Parameters page, I check those, click Next, and on the Finalizing Restoration page it does not complete, as per the screenshot.

The error_log in the installation folder has this:
[09-Jun-2016 11:11:37 America/Chicago] PHP Warning: require_once(1/platform.php): failed to open stream: No such file or directory in /home/bitlinkh/public_html/clients/Luther5/libraries/import.legacy.php on line 42
[09-Jun-2016 11:11:37 America/Chicago] PHP Fatal error: require_once(): Failed opening required '1/platform.php' (include_path='.:/opt/alt/php56/usr/share/pear:/opt/alt/php56/usr/share/php') in /home/bitlinkh/public_html/clients/Luther5/libraries/import.legacy.php on line 42