Thanks a lot, Usman, for all your help.
It is very interesting that when I installed PHP 7.0.2 on my Moodle 2.9.1 I can see certain pages OK but certain pages are simply look blank (nothing is displayed). That is why I suspect that some plugins that I use are incompatible with PHP 7. Unfortunately, backup is one of these blank pages.
PS Also, I've messed with php scripts on some pages to make our Moodle more dialogic, democratic, and collaborative for our classes, so it will be a lot of "fun" to adjust these scripts to Moodle 3.x and PHP 7 in the summer for me . But the backup php page is not affected by my modified scripts, so it should be plugins on that page that are incompatible with PHP 7...