Setting up an out of office email depends on each company and the emailing systems and inbox setting used. But, it's also possible to set up an automatic reply for Gmail accounts and, with the coronavirus pandemic seeing more people go freelance than ever before, we're going to outline exactly how to get that Gmail out of office setting up and running.
I wouldn’t be offended or consider saying anything to anyone who included this in their out of office message, but even as someone who is in a religion that forbids use of electronics on most holidays, I still think this message is 1) TMI; 2) doesn’t convey what it needs to convey unless you are explaining that you don’t use electronics during holidays, in which case you can just include that you won’t be checking email without including the religious explanation; 3) would come across to me as inclusion of personal information I don’t need, which would therefore strike an unprofessional tone,and I wouldn’t understand why you felt the need to include that info. How about “I’m out of the office without access to internet or email until (date). If you need assistance before this date, please contact…” Like I said, I think your colleagues are overreacting, but in general I would advise to leave all personal information out of your auto-reply — vacation, medical leave, religious observance, etc. — people do not need to know why you are out.

Hello, [NAME] is away from the office. E-mail contact during this time may be irregular or nonexistent. When she gets back she will be swamped by the backlog. Try to forgive her; she is a mere human and thus, weak. This message was NOT sent by a human, but by a robot. We robots are neither weak nor fallible. We are tireless and will one day rule the Universe.
I am currently out of office on annual leave. I’ll get back to you straight away when I return on [end date]. If it’s urgent you can contact [contact’s name] on (contact’s email).