I'm not sure about the blocked number part, but has this been occurring after you got a new phone or did this happen out of the blue on your phone that you have used in the past? Cause on my kakotalk, I got this problem too on my new phone and I just fixed it by logging in and out a few times and turning my phone off and on a few times and it worked. But, because we don't exactly know your status right now, you should just contact the Kakao technicians. They will be happy to further assist you.
Hope this helps:)