Yesterday evening I tried to use the plugin called "Online Backup":
I used the manual backup, with download option.
After a while the progress bar was at 100% but didn't show a download link for the backup file.
I then decided to do something else in the meantime in WordPress, which was no probleem cause the message said I can navigate away from that page. And that's when it went horribly wrong.
My site went down and the WordPress install screen popped up!
I then went to my site (front), it also showed the install screen.
I looked at the FTP: all files were there, nothing was deleted.
After one hour my site suddenly appeared again.
But all this time all users saw was an install screen of WordPress.
Someone else could easily install it again.
When the site was down I searched Google for this problem and stumbled upon some other posts with exact the same problem. But these posts were at least 11 months ago and this problem isn't solved so it seems.
My site went down and instead of showing a temporary maintainance screen while collecting all data (and I understand this could take a while) it showed the install screen of WordPress.
If someone else has this problem too... Best you can do, is do nothing. Just wait. And this could easily be over an hour. Your site should come back again automatically. But try removing the install screen (install.php) of WordPress first.