Our store will be closed until the end of the week for [Thanksgiving/Christmas/New Year]. We are happy to inform you that all of your emails will be answered once we are back on [date].
Once the person reaching out to you realizes you’re gone, they may panic if they have a legitimate need for immediate support. The best way to avoid this is to let your callers (and emailers) know how to find the appropriate help.

We had someone today that sent an inquiry about something (this person is not a client, more of an outside business partner). Dude sent, within about an hour and a half time frame, 7 emails. Calling out one person on the DL multiple times within a half hour, then proceeding to call out the rest of the DLs (ALL in the original copy list) to try to get an answer to his question.
There’s a grim, apologetic vibe to these messages — I’m sorry I’m taking time for myself but I’ll try to check in on occasion! They’re a vivid reflection of a work culture that valorizes constant productivity and the near-total overlap of work and life. But they’re also do a terrible job of what they’re intended to do, e.g., set realistic expectations for both sender and recipient. A vague OOO message traps both parties in an uncomfortable liminal space where both productivity and rest go to die. The original sender is left unsure if they’ll be getting a timely response or a whether the email will go ignored for a time or forever. The original recipient has taken what is a rock solid excuse (time off) and cheapened it, offering a backdoor for email guilt to creep in.