Yes, mine (for external e-mails) typically says somethingalong the lines of “I’m out of the office until [date/time] and messages to this address are not monitored in my absence. I will respond as soon as possible on my return. If your message is urgent, please re-send to my assistant [email address] or telephone [assistant’s number]” Internal it will usually just say “I’m out until [date/time] pass any urgent enquiries to [co-workers] ” although my assistant and a couple of others have my home number and personal e-mail so can get hold of me in a genuine emergency, and I will sometimes speak to them in advance if I’m willing to lower the bar on what amounts to an emergency, but I would not expect any of my employees to do that !
An out of office message shows your professionalism by informing others of your unavailability. Such messages generally include an apology for the inconvenience, a reason for not replying right away, the time the receiver can expect a response and an alternative person they can contact if there is something urgent.

I worked for a federal contractor back during the Great Recession when government offices were shut down/working with a skeleton crew. I still remember getting OOOs from almost every email address in the agency we worked at explaining they were on furlough & to contact one specific person if the issue was urgent. We all assumed this poor person was hiding under her desk, rocking back & forth, with her head in her hands.
It’s important to get the tone and content right because it can make or break your reputation as a company.