This had me stumped for a while, so I’m posting this for other, and to jog my memory if I forget.
I’m not sure when it happened, but in more recent versions of WordPress the page/post editing screen has been cleaned up by default to not include some of the (presumably lesser-used) options.
The way to re-enable them is via the “Screen Options” menu in the top right-hand corner:
Once clicked, it opens a menu where you can toggle various aspects; Categories, Tags, Excerpt, Send Trackbacks, Custom Fields, Discussion, Comments, Slug and Author.
Discussion is the option you need to turn on to disable comments, as “Comments” only shows the existing comments. Somewhat disconcertingly, I found that the Discussion section, when it appeared, was empty:
However, if you click on the Discussion bar, it will expand and show you the check boxes for “Allow comments” and “Allow trackbacks and pingbacks on this page”.
Not all that intuitive, I guess. But you may find similar features hidden in the “Screen Options” on other parts of the admin dashboard. Check it out.