We're pretty much surrounded by internet wherever we go these days but still, I could imagine taking four weeks off without any access to the internet and wouldn't want people to think I died.
You'd need to only answer every email 27 days after receipt, which can be easily automated (no the answer, but you could delay it being shown in your inbox, or fowarded to your main or something) but it would be a pain to keep it up and not reveal info that could be harmful to the still-living you.
and when you die... spam everywhere? Is that really how you'd like to be remembered? :)