Tried to break this sentence down, but I had trouble.
번호 is number, but I tried to look up 거신 / 거시다 but didn't get very far. So I'm not sure where the "dialed" part comes from.
Nevertheless, if we assume (지금 거신 번호는) to be the subject ("just-now-dialed-number"), the next part is 없는 번호입니다 which, to me, means "is a number without". Does this mean "number not in service"?
Answering my own question 8 months later (with the aid of the below reply).
"지금 거신 번호는 없는 번호입니다" broken down:
지금 거신 번호 = (literally) The number that you have just called / dialed
거신 is honorific here; it comes from 걸다 + 시 + ㄴ (past tense modifier). 걸다 has many definitions, and among them, "to dial/call" is one of those definitions.
없는 번호입니다 = It is a number that does not exist.
Put together, and the result is "The number you have dialed does not exist", or in other words, "The number you have dialed is not in service."