Data recovery from Mac computer that is failing to boot up

First thing I would do is check the health of the drive as that will ultimately determine your options for data recovery and also the likelihood of success. More than likely the hard drive is bad which is why you cannot boot your old iMac.

The easiest way to test the iMac and the hard drive is by creating & booting a Knoppix Linux USB stick. If Knoppix boots, then most likely the internal hard drive has failed. It is possible to use Knoppix to pull the health information from the drive which you can post here so I can examine the report to see how bad the failure is. See my other post here with instructions for creating & using a Knoppix Linux USB stick as well as directions for access the hard drive's health information:

Creating & booting Knoppix Linux USB Stick to check drive health -- HWTech - Apple Community

If the drive is failing, then most regular tools & utilities will be unable to recover your files since they cannot handle all the errors produced by a failing hard drive. The more you try to access a failing hard drive, the worse the failure becomes where even an expensive professional data recovery service will be unable to recover any data.

Another Hail Mary is to try Target Disk Mode:

Transfer files between two Mac computers using target disk mode - Apple Support (CA)

Bot from another drive & try Disk Drill...

https://www.cleverfiles.com

Do you have another Mac? If so which exact Mac?

No the Brick and Mortar Apple Stores will not service or work on older Mac's.

You will need to call and ask your local Apple Authorized Service Provider's to see if any offer data recovery.

to locate your local AASP's, go to > https://getsupport.apple.com/repair-locations?

Keith Barkley wrote:

Another Hail Mary is to try Target Disk Mode:

Transfer files between two Mac computers using target disk mode - Apple Support (CA)

No, Apple Stores do not offer data recovery. Your data is likely gone forever, this is why Apple introduced Time Machine almost 20 years ago, to back up your computer. If somehow you are not aware of Time Machine please click the following link so this does not happen to you again.

Back up your Mac with Time Machine. Time Machine only requires a simple External Hard Disk such as a 2TB OWC Mercury Elite Pro.

Note: Operating ANY computer without a sound backup plan is an almost certain guarantee the user will lose data. I know this is not what you wanted to read and I know it is painful to lose data (it's happened to almost all experienced users including myself). When you lose data, the upside is hopefully you will have learned a lesson about why backing up is absolutely critical.

By the way, data recovery services are available however they are extremely expensive and also cannot guarantee exactly what data they can recover if any.

Best of luck to you!

If you do not have a backup using Time Machine, CarbonCopyCloner or SuperDuper on an external drive..?

Then your only option is to use the Data Recovery Service and hope that they can Recover all of the Data from the drive.