Size of a MAC for a quickly checked message? Let suppose that we have to check a message that was written one second ago. The message is discarded immediately after having being checked. What "minimal" size for such a MAC is secure?
If the MAC is well designed and there is no better attack than guessing, the attacker can choose at least two attacks:

a. Find the key for MAC and be able to sign any message.

b. Guess (maybe randomly) a MAC of an arbitrary message. (If unsuccessful, maybe retry the attack.)

The feasibility of attack a depends on the size of the key, but not on the size of the MAC itself. (Well, smaller MAC means potentially less information leaked about the key, but this is rather a minor effect.) Small MAC thus does not increase the probability of the key being cracked.

The important point: Attacker without the key can't just bruteforce a MAC for an arbitrary message, even if the MAC length is 1 bit. Since MAC is symmetric and key is kept in secret (there is no public key), attacker with a valid pair of message and its MAC can't verify it offline.

None of these facts, however, mitigates attack b. The attack b may be a real threat with a short MAC. Feasibility of the attack will depend on details that are not available. (It, however, does not much depend on the time window between message sent and message recieved.) If the MAC does not protect anything very valuable and the attacker is likely to be traced and killed after the first unsuccessful attempt to forge a message, one bit MAC might be very sufficient. If the attacker can do many unsuccessful tries without any problem, then the MAC must be much longer in order to resist such attack. How much long? It depends on the feasible online attack rate. Plus, you are likely to want a safety margin, don't forget about that.

There is one more issue. Where would you get a MAC function that outputs a shorter MAC? When you take an obsolete MAC function, there might be some serious issues (e.g. short key). When you use an existing MAC function and shorten the output from it, it is not guaranteed to provide desired level of security. For example, a MAC function with 128b  output might provide signifficanty less enthrophy in the output due to some redundancy or probability characteristics. This should not be an issue with HMAC, which is hash-based.

I hope you question is rather theoretical one (i.e. you are curious if this is safe) and you don't wan't to implement such system. The MAC overhead should be usually negligible and there should be no serious reason for using a short MAC. The benefits of slightly smaller MACs are usually not worth of a serious reasoning about security of that system.