Yes, it’s possible, with this workaround currently: This module allows to limit user with ‘edit_others_posts’ capability to edit just posts from the list of authors – field “with author user ID”. If you input into this field the ID of user himself, he will see in the posts list just his own post.
I think to add direct checkbox ‘Own posts only’ to one of a future versions.
Starting from the next version 4.25 special setting ‘Own data only’ will be available as for a user, as for a role. It will be no need in a workaround described above.
Development version 4.25.b7 is available after login from the Download page. It’s under testing currently.