Not that I'm aware of. You'd need to run a reverse proxy somewhere and then point it to your home server as a default but having your other server as a backup running the software at the same time. The big problem is how to store the data on both servers at the same time, especially small texts which are in the database.
Thinking out loud here, but if this is public-facing you can put together a static archive using something like ArchiveBox/SingleFile and have the monitoring server serve the files and update the DNS entry when it detects downtime past a certain threshold.
Sure, it won't offer exactly the same affordance but the OP mentioned rare downtimes like electricity going out (rare here is used generously, obviously some places see this happen often).
Providing a read-only, static version of the services, particularly toots and blog posts, while the origin is unreachable is straightforward and inexpensive.
Somewhat related, the Solar Protocol [0] does something like this to host websites across an array of solar-powered servers across the globe.
http://solarprotocol.net/
