For a computer that old I would definitely first suspect a failing hard drive. It was also my first instinct when I first replied to you. Plus the issue with the iMac got worse after Apple started working the drive harder while attempting to fix issues because Apple's support techs do not know how to check the health of a hard drive (Apple assumes users have a backup and doesn't care if things get worse because the backup can be used to recover everything, so all their support instructions neglect to verify the health of the drive).
Personally I would first check the health of the hard drive. The severity of the failure determines the best course of action for recovering data from the iMac. Keep in mind the more a failing drive is accessed or even powered on, the worse the failure can become where even a professional data recovery service may be unable to recover any data from it.
Assuming the drive is healthy or is only just beginning to fail, then I would suggest installing macOS 10.11 to an external drive so you can boot the iMac using the external drive. If the internal drive is just beginning to fail, then it may be possible to transfer most of the files to the external drive (there are a couple of ways to do this depending on the health of the internal drive -- either manually through the Finder or using Carbon Copy Cloner the latter is able to skip files producing errors and log which ones could not be transferred). I would probably do this even if the internal drive is healthy since you never know if First Aid could make the problem worse and lose access to the data (probably not, but there is always a risk if you don't have a backup). Plus if the internal drive is physcially healthy and First Aid is unable to fix the file system on the internal drive, then you can attempt to repair it using a third party utility that tends to work magic on corrupted HFS+ file systems.
If you have access to another Mac, then you could attempt to access the data on the iMac's drive by putting the iMac into Target Disk Mode. Again, you should only attempt this if the hard drive is verified to be healthy or only has a few bad sectors so far.
Checking the drive's health requires using a special utility to access the hard drive's built in health information to get an accurate idea of how bad the failure is. The Apple Diagnostics will only tell you the drive has issues, but not how severe the issues are plus the diagnostics don't detect most drive failures. The special utility to access the hard drive's health information is the only way to get a reasonably accurate idea of the drive's health.
