I am trying to make a backup of my new Linux system.
I created a SystemRescueCD and booted to it in order to run FSarchiver.
How do I run FSarchiver to backup the Linux system on /dev/sda7 to a newly formatted LinuxBU partition at /dev/sdb3 ? Do I need to mount /dev/sdb3 some how, but not mount /dev/sda7 ?

Good that you are doing backups - better if you had read the manpage so you understand the tool. There is also a quickstart on their web site.
fsarchiver produces a single file - so the target has to be mounted. The source is a device node - that implies unmounted.

I read the manpage and the quick start guide. Neither was much help. It seemed like the destination needed to be mounted, and the source file system unmounted (the documentation was not particularly clear). I mounted the backup disk with
mount /dev/sdb3 /mnt/backup
and then, from SystemRescureCD /usr/sbin ran
fsarchiver-64bit -v saved /mnt/backup/Linux249.fsa /dev/sda7
and got an error that /dev/sda7 was not a block device. Could this be due to my installing Linux in a logical partition within an extended partition?

Now what?

When you are told that something "is not a device", your first step should be to look for it in the /dev directory. Also check that it has a type of "b" for "block". For example, here is my /dev/sda7:

brw-rw---- 1 root disk 8, 7 May 15 06:45 /dev/sda7

If it isn't there, run dmesg|grep sda and see what the kernel did with that drive?

That's not a valid command - you should have gotten a screen full of messages.
If systemrescue has a standard build (other than the name) you want savefs - try thisAnd no, the fact that it is a logical partition makes no difference at all.