I don’t want to do any sort of RAID 0 or striping because the hard drives are old and I don’t want a single one of them failing to make the entire backup unrecoverable.
This will happen in any case unless you had enough capacity for redundancy.
What is in this 4TB drive? A Linux installation? A bunch of user data? Both? What kind of data?
The first step to this is to separate your concerns. If you had e.g. a 20GiB Linux install, 10GiB of loose home files, 1TiB of Movies, 500GiB of photos, 1TiB of games and 500GiB of Music for example, you could back each of those up separately onto separate drives.
Now, it’s likely that you’d still have more data of one category than what fits on your largest external drive (movies are a likely candidate).
For this purpose, I use git-annex.branchable.com. It’s a beast to get into and set up properly with plenty of footguns attached but it was designed to solve issues like this elegantly.
One of the most important things it does is separate file content from file metadata; making metadata available in all locations (“repos”) while data can be present in only a subset, thereby achieving distributed storage. I.e. you could have 4TiB of file contents distributed over a bunch of 500GiB drives but in each one of those repos you’d have the full file tree available (metadata of all files + content of present files) allowing you to manage your files in any place without having all the contents present (or even any). It’s quite magical.
Once configured properly, you can simply attach a drive, clone the git repo onto it and then run a git annex sync --content and it’ll fill that drive up with as much content as it can or until each “file”'s numcopies or other configured constraints are reached.