I have my e-mail hosted externally at a place called aiso who advertises they are a solar powered data center. So they should stay up for a while.

Your last mile connection to the internet depends on who you have for your local utility. For example local phone service here was bought out by AT$T a while back so you have maybe 50% reliability even when there is no disaster. We switched to a local cable company who keeps the service running and I've user the power companies web site to report my power being out.

The original context of the question was around access to important data. I use the internet to sync data to and from my phone/tablet/laptop but run my own server at home. Since my data is a copy on all three devices and the server its accessible from any of those weather they can get a connection or not. The network connection simply keeps changes in sync automatically so if I update a phone number or something I don't have to manually copy the doc to all my devices.