EtreCheck says my hard disk is failing--should I believe it and just by another computer?

You could first try First Aid on Disk Utility: https://support.apple.com/en-us/HT210898

Failing hard drive - This machine has a hard drive that appears to be failing.

It isn't always failing when you see that message, could be etrecheck thinks something is wrong with it just due to so little free RAM & so much swapping going on, slowing the already slow drive like crazy.

Delete these files ^ restart...

/Library/LaunchAgents/com.teamviewer.teamviewer.plist

/Library/LaunchDaemons/com.teamviewer.teamviewer_service.plist

/Library/LaunchAgents/com.teamviewer.teamviewer_desktop.plist

~/Library/LaunchAgents/com.adobe.ARM.***.plist

~/Library/LaunchAgents/com.yahoo.YahooContactSyncAgent.plist

~ tilde indicates hidden Home Library.

Apple hid the Users' Library folders...

Method 1:

1. From the Finder, select the Go menu at top of the screen, and choose Go to Folder.
2. In the window that opens, enter ~/Library, and click Go.

Method 2:

1. Go to the Finder (or desktop).
2. Hold the Option key on your keyboard, and click the Go menu at the top of the screen.
3. With the Go menu open, you'll notice that pressing and releasing Option will display or hide the Library choice in this menu.

 4 Select Library from the Go menu (while holding down Option) to access the hidden folder.

Disconnect your external drive(s) and retest since one of those drives could be interfering.

I would be concerned about the Seagate & WD proprietary software as they have been known to cause problems with macOS. At the very least make sure they are completely up to date. Even better would be to uninstall them.

To check the health of the hard drive(s) run DriveDX and post the report(s) here using the "Additional Text" icon which looks like a piece of paper. If you want to check the health of the external drives you will need to install the special USB driver. Even with the special USB driver there is no guarantee that the external drive's USB controller will allow the necessary communication to occur.

First off: assume Etrecheck is right, and make sure you have an up-to-date back top at all times. If the HD is failing you could very quickly find yourself unable to start the machine and unable to recover the contents of the drive. Take that warning very seriously.

Then get the machine checked by an Apple Store or Approved service centre. You can't fix hardware problems with software. If the HD is failing then it can be replaced. Frankly, there's so much at stake here that I would only let a professional look at it.

Like somebody else wrote, take iMac to Apple Store, or Authorized Apple Service Provider. So they can find out what’s really going on. They probraly go charge you some “diagnostic fee,” then go from there.

This:

System Software:

macOS Catalina 10.15.5 (19F101)

Time since boot: About 7 days

Try to restart the computer every 2-3 days. With only 8GB of RAM that is not user-upgradeable, more frequent restarts will take some load off by reducing the writes from virtual memory to the already-slow hard drive.

Next: DivX is over seven years old. That's Jurassic in computer years. There is evolving evidence that older DivX versions may slow newer macOS systems like yours. If you use it, upgrade it; otherwise dump it.

+1 on HWTech's concert about your external drives. Remove the WD and Seagate software. I've had slowing, stalls, and overheating when running WD externals with their proprietary software. WD bare drives are fine--it is their cheap enclosure hardware that is troublesome. I've pulled the drive for inside a WD enclosure and placed it in an OWC enclosure. The drive required no additional software and ran fine inside a quality enclosure.

As it looks like the drive may take the long trail any day, make sure you keep your Time Machine backups current. Verify that entering Time Machine allows full access to file; don't assume it is working.

Consider a secondary bootable backup using Carbon Copy Cloner to a different external drive.

You have some very good comments here. I will add a couple. Yes, your RAM is too little. I believe that your computer is one which you can not add RAM to by adding modules. Rather than replacing the hard drive, I suggest (if you decide to keep tis computer) purchasing a solid state, SSD, external drive and then installing an OS on it. You can use Migration Assistant to transfer your data to the new drive. It should run a bit faster than your old hard drive. To help things, restart your computer regularly, clean out the browser(s) history periodically as this also uses RAM and do not keep open apps you are not currently using. These steps help reduce the RAM's workload.

Last, if you do buy a new computer, be sure to get one with an SSD and no less than 16 GB of RAM, preferably more. I would get the one with the larger screen as these units allow RAM to be easily upgraded by you.m Good luck to you.

Your iMac has slow 5400 RPM hardrive, not enough RAM, only 8 gigs. 32 bit apps will not run on MacOS Catalina, need 64 bit apps.