I was using an external hard drive this morning and mistakenly disconnected it physically before I had unmounted it. Later this afternoon I tried connecting the hard drive and it’s not mounting. Then it started making some squishy notes.
Uh oh!
I tried another cable, and also connecting the drive to a different device altogether, without success.
Any ideas? Has the drive just died?
Try connecting the drive to your computer and reboot. If it doesn’t mount, open Disk Utility and see if you can get it to mount from there. Because your drive was making “squishy” noises, if it does mount, I would copy all files off of that drive.
I tried this:
The drive still doesn’t mount 
Did you try to mount through Disk Utility?
I did. The drive doesn’t appear on Disk Utility, nor on the Finder, and therefore I can’t mount it. Unless I’m missing something?
If there’s a problem reading the drive, it can take a much longer time for a drive to mount. If you haven’t done so already, maybe try leaving it plugged in for a while? I’ve seen corrupted drives that didn’t look/sound like they were doing anything mount after 10-20 minutes.
Also, not sure how comfortable you are with Terminal, but this might be a job for the command line? e.g. https://www.applegazette.com/mac/pro-terminal-commands-using-diskutil/
Obvs, if the command line isn’t your thing, try everything/anything else before you go down that path, and make sure you’ve got some knowledgeable support at hand if you do. The risk of inadvertently hosing something else on your system is too great, otherwise… (insert standard responsibility disclaimer here…)
If you have just about given up BUT are ready to try one last thing…
Put it in the freezer overnight. Seriously. Take it out and as @jsamlarose says
if it doesn’t mount, try the terminal and then do what @neonate says and
copy everything off it immediately. Best of Luck!
My understanding is that the freezer thing doesn’t really work for modern hard drives. I did try it, though it didn’t make a difference.
Ultimately, I bought a new hard drive and subscribed to cloud backup service (Backblaze).
The new hard drive failed within a month or two. I got it replaced and was able to recover the data from Backblaze.
All of which is to say, you need two sources to have a backup, otherwise you just have an archive 
Agree completely. However, at no point was there any discussion of
how “modern” the hard drive was. 
Glad you got it sorted, and your guidance on backups is spot on.