My upgrade from Mojave to Monterey failed. What does this error message mean "Storage system verify or repair failed." (-69716) ?

I would try incrementally upgrade. First go to Catalina . If successful go to Big Sur, If successful then to Monterey.

You can get Catalonia and Big Sur here:

How to get old versions of macOS - Apple Support

Do you have a backup of your data? I would start there.

Upgrading from an older macOS version such as Mojave straight to Monterey is not recommend, you will be skipping Catalina and Big Sur in the process and there were a lot of changes in those two releases.

The error indicates a potential disk issue either hardware or filesystem related. Reformatting the drive may be necessary and having a backup of important data and files would be necessary as a reformat will erase all data.

Depending on if you have a lot of long term data and apps installed you might want to consider upgrading from Mojave to Catalina patching it fully then upgrade Catalina to Big Sur also patching it fully then upgrade to Monterey.

The alternative would be to completely erase the iMac and install Monterey with an external thumb drive cleanly. Then re-install your Applications and copy your data from your backup.

Here's some support articles:

How to get old versions of macOS - Apple Support

How to create a bootable installer for macOS - Apple Support

Do you have a Fusion drive? That is a 2TB HDD and 1020GB or so SSD?

If so there are problem upgrading to Monterey with a Fusion drive.

Maybe these will help:

Cant Update software - “storage system ve… - Apple CommunitMacOS

Mojave to Monterey - Storage System… - Apple Community

The Age of this computer will Govern what newer version of macOS it will Qualify to install. Use About this Mac from the Apple Icon on Desktop - the required Information will be there.

With the above information in-hand - follow this How to get old versions of macOS and verify what version this computer Qualifies to install.  For Best results use Safari to commence the download as Others may not work.

Safari will open a redirect and open the Apple Store where the download can commence.

If one gets an Error message in the Apple Store or no ”  Get Button “  Replace the https in the page's address to macappstore. Credit to Contributor @Niel

Computer Qualify for the Monterey upgrade

macOS Big Sur 11 hardware requirements

macOS Catalina 10.15 hardware requirements

Caution - jumping over all the newer versions and directly to Big Sur  / Monterey on Qualifying computers is Technically possible - there are those who have attempted and were successful. Then there are those who attempted and had less than good results.

If this is a Production or Mission Critical Set-up - a more prudent and measure approach is suggested. 

The above is from observations on the Apple Support Community Forums and IMHO

Tips Big Sir Upgrade Released Nov 2020

Tips Monterey Upgrade  Released Oct 25, 2021

Make Time Machine Backup  is very useful and can be used to Revert to Previous Working macOS

Notation - Mojave is the Last version to support 32 bit applications / extensions and Drivers. Catalina, Big Sur & Monterey supports only 64 bit and there is no workaround.

Suggest using Go64 to find 32 bit application before attempting the upGrade. Some or many of the Application already installed may required update or upGrades to 64 Bit Versions.

The iMac is about 7 years old so it's probably in the last year or two of it not being able to run what comes after Monterey. You need to consider the costs of fixing this and keeping it going until you can buy a newer one.

Proposed plan of action:

1. BACKUP YOUR DATA (Recommend a full clone backup using Carbon Copy Cloner or SuperDuper)
2. VERIFY YOUR BACKUP IS GOOD
3. Create a bootable USB flash drive with Monterey installer and boot from the USB by holding Option key at power on.
4. Completely erase the internal drives the 120GB SSD & the 2TB SSD with Disk Utility (show all devices)
5. Format the SSD and HDD internal drives with APFS and run First Aid on them starting at the volumes working up to the containers and finally the hardware itself.
6. Open up Terminal and execute the command "diskutil resetfusion" this will rebuild the fusion pair of the two drives. All data will be destroyed.
7. Then run the Monterey installer onto the newly created Fusion drive.
8. Re-install your apps and restore your data from the cloned backup by dragging and dropping what you need.

Assuming you don't find a disk failing this should work.

If you do have a failed SSD or HDD then the question will be if it is worth it to find a local shop that can safely open the iMac, replace the fusion drive pair with a single large SSD while factoring in labor and part costs. Doing this yourself is possible but it can be risky as it's easy to break something. You'd need to buy some specialized tools and watch some details videos, etc. Even professionals would struggle to do this the very first time. Most would be nervous about this procedure.

Using a 3rd party SSD will mean first installing Monterey on the original SSD to ensure the Macs firmware is updated. Lots of MacBook Pro users with replaceable SSD's found this out the hard way. They couldn't upgrade to Monterey because the internal SSD wasn't an Apple one. They had to remove their 3rd party SSD, put the original Apple one back in. Install Monterey and patch it fully. Then remove the Apple SSD and put the 3rd party one back in and only then would Monterey install successfully.

Think about the time and energy this will take plus the cost in labor and parts for a Mac that might only last another year or two before it won't run the latest macOS and the year or two after that you start running into software that requires the newer macOS version you can no longer run.

You are looking at perhaps getting things to last long enough to be ready when the newer iMac models drop some time this year. Or perhaps the M1 iMac 24" would be fine. If so, it's a great Mac and it will definitely be many times faster than what you have now. I would avoid the entry-level model and go for the mid-range to higher-end model. Otherwise you won't have a cooling fan and extra USB3 ports plus Ethernet on the power supply.

Open Disk Utility and go to the View menu and click Show All Devices. Take a screenshot and post in reply.

Open Terminal and issue the following command and attach the output.txt file which you will find on your desktop using the additional text button in reply.

diskutil list > Desktop/output.txt; diskutil apfs list >> Desktop/output.txt

Do you have a Fusion Drive? A pairing of an SSD and an HDD. Apple offered this option on iMacs for many years but there is a problem if either the SSD or HDD starts to experience a hardware problem.