What excuse would you use?

[grit]

I could be wrong but I think if you have executive responsibility to a board and staff, you must have emergency number that anyone could reach you provided it is emergency. No one with responsibility of others' livelihood can go hidden anywhere on earth - that is one of the reasons some people do not want to become managers or higher. However, you have privacy rights and can have a family member who could answer your phone calls or keep track of what needs your attention in case there is a real emergency like your office building is on fire or death at work etc.
All I can think of is you could go on silent retreat. No one will expect you will speak or type and it is so common these days. If you want to make it more interesting, you could put it a place that has 12hrs time difference.
Remember that Governor who went to the Appalachians hiking and was found having affairs...try to avoid situation like that.

 

[Cypress]

I agree with the earlier posts, something vaguely spiritual like a silent retreat in the mountains, no details needed. I also agree that giving an emergency number to your appointed lead will be important in making everyone feel more comfortable. The emergency number can be a friend or family that knows what is going on and can pass on the message so that you can call back when you are prepared.

Is there someone on the board you trust and that won't balk at being in an advisory role that can act as a liaison with your staff if they really need it?

Try to spin it this way: it will be a great opportunity for your staff to develop better leadership skills.