So, I did it. I unplugged from all my computational resources for 24 hours this weekend as part of International Shutdown Day. What was it like? Peaceful.
I enjoyed it so much, in fact, that henceforth I am adopting a new email strategy: I'm only going to check email twice a day, once in the morning and once in the evening. I'll process all the messages, clear out my inbox, and then shut down my mail client until the next schedule check. This is an experiment and may not last the week, but the freedom I felt while "unplugged" this weekend was really remarkable, so it's worth a try.
Note to those who would send me emails: if it takes me up to, say, 12 hours to reply, don't fear for my safety. I'm fine - I'm just actually something productive with my time.