I had a rather large Akeeba Backup file (about 25 GB) which I tried to restore using Akeeba kickstart from the web browser. It took a long time unpacking the archive but never reached the point where database is being restored.
The Akeeba kickstart (version 5.0 or later) can be run from command line (shell) and this works better with really large backup files. Simply go to the directory where you have your JPA-file which should be the same directory where it should be restored and enter:
php kickstart.php archive.jpa
But of course replace archive.jpa whith your archive name. There are more command line options available.
After unpacking the archive, go with your browser to https://yourdomain.com/installation and complete the restore.