I don't believe that anyone is born unlovable and I believe that love is something others 'give', I suppose, rather than there being something that we do to elicit it. Maybe. I don't know!
But can people be so damaged by their life experiences that they become unlovable? That people can't see lovable qualities in them?
Even when people have come to a point through therapy or choice to 'love themselves' - confident, good self care, good boundaries, treat themselves kindly and with respect, can there still be 'something about them' that makes them unlovable to others?