Add comments to 404.php page

Is it possible to add a comment block to the 404.php page? I’ve tried to add the basic <?php comments_template(); ?> to the page but the comment block doesn’t show.

The reason why is I’ve got a rash of 404’s that from Google Analytics seems to be for a specific link “/SG” and appears to be coming from users landing on random pages on our site (http://www.comtrya.com) then clicking on this bad link. When I check those pages this link doesn’t exist. So I want to ask users who land on the 404 page to leave a comment about what the clicked on to get there.

Any help or thoughts on this appreciated, David