I work for a hospital, in a role unrelated to patient care. My first out of the office message was just my name and department. After a series of increasingly plaintive messages one evening, I added, “If you are calling about patient care, you have the wrong number.”
That said, I think it is the kind of thing that is funny with the right people and in the right situation. But an out of office message is an autosend situation, so the email system cannot actually assess if it is appropriate or if the person receiving it will find it amusing, or unprofessional, or apparently even condescending. So while it is a hilarious message for a joke, it would not be a good idea in a professional setting!

Give yourself some slack when promising people to keep up with their messages. If your vacation ends on January 18, but you know that you won’t be able to check up on old emails for the next couple of days, mention that in your reply.
There was a lot of pushback on this, discussion about how big a risk was that really anyway, people saying that John could word his out of office in such a way that people didn’t have to know he was actually away, and if something really was an emergency people would like to know that they have the opportunity to “direct queries to Sam or Dean” so they could be actioned, or make the judgement call that something could wait for John to return.