If a barber accidentally gives someone a bad haircut, causing them to miss a meeting and contract, can the person sue for damages?
Yes, they could sue either for breach of contract or under the tort of negligence. If they succeed, they would be entitled to either a) a refund or b) another haircut.
They would not be entitled to damages for "miss a meeting and contract" as these are not foreseeable results of a bad haircut. That is, the barber's negligence is not sufficiently proximate to the loss.