Agreed, ZFS does exactly this, though without the remote file retrieval portion.
End-to-end file system checksumming is built in, but by itself this will only tell you the files are corrupt. To get the automatic correction, you also need to use one of the RAID-Z modes (multiple drives in a software raid). OP said they wanted to avoid that, but for this kind of data I think it should be done. Having both RAID and an offsite copy is the best course.
You could combine it with some scripts inside a storage appliance (or old PC) using something like Nas4Free (http://www.nas4free.org/), but I'm not sure what it has "out of the box" for doing something like the remote file retrieval. What it would give is the drive health checks that OP was talking about; this can be done with both S.M.A.R.T. info and emailing error reports every time the system does a scrub of the data (which can be scheduled).
Building something like this may cost a bit more than for just an external drive, but for this kind of irreplaceable data it is worth it. A small atom server board with 3-4 drives attached would be plenty, would take minimal power, and would allow access to the data from anywhere (for automated offsite backup pushes, viewing files from other devices in the house, etc).
I run a nas4free box at home with RAID-Z3 and have been very happy with the capabilities. In this configuration you can lose 3 drives completely and not lose any data.