iTunes says I can't back up iPhone because I don't have enough space...but I do
So after much swearing and several hours of fiddling, I finally discovered what the issue was.

Windows appears to have a limit on how many files can exist in a directory tree. iOS backups consist of tens of thousands of individual files. And if you have several backups in your backup folder, we're talking a whole lot of files.

The solution ended up being fairly simple:


find your backup folder on your hard drive
move most (or all) of the backups that are in that folder to a separate folder and/or drive.


Then try backing up again. For me, that solved it and I was able to get the backup to run.