IMAC (27-INCH, MID 2011) won't complete start up- just white screen,black screen or reboot loops

The first thing to do is make sure you have a current and full backup of your hard drive.

Then you might try boot into the Recovery volume (boot with the Command + R keys held down), select Disk Utility and repair the disk. If it's able to do so try rebooting and see if that will let you continue with it. If it does't try booting into the Recovery volume again and reinstall the system. Reboot and keep your fingers crossed: 🤞

If the above fails try booting it up into Target Disk Mode.  If it can then the problem is not the hard drive but could be either software or other hardware related. If it can't boot into Target Disk Mode then the drive is probably trashed. You could try getting an eternal SSD to boot from via Thunderbolt but that can be rather costly and of questionable cost effectiveness on such an old machine.

Since that model is considered obsolete by Apple trying to repair it will be difficult. You'd need to find an Apple Authorized Service Provider who is willing to have a go at it but would be costly in labor costs alone. It might be time to consider getting a new iMac.

I had a 2011 27" iMac that I removed the hard drive from. I had both a regular HDD and an SSD but separate, not a fusion drive. The screen pops off as it's only held by magnets. The HDD was rather easy to find and remove but the SSD was behind the motherboard and took some "tearing" down to get to. I took both of them out, jammed the remaining parts back in and put the screen back on before taking it to Apple for recycling.

I believe this link will show you how to get at them: Installing iMac Intel 27" EMC 2390 Dual HDD or SSD Drive - Installing a secondary HDD or SSD in the mid 2010 27" iMac EMC 2390.

or this one: iMac Intel 27" EMC 2390 Hard Drive Replacement - Replace the hard drive in your iMac Intel 27" EMC 2390.

Susan, it sounds like a hardware failure however we cannot be sure on a forum. Please try resetting the NVRAM and restarting in Safe Mode. If neither work then you can take it to your local independent Apple Authorized Service Provider (Apple Stores will not work on it due to it's age). At least then you can find if if it can be repaired and what the cost would be.

However, considering it's age, I would not spend money on it, I'd invest in a new iMac.

How to reset NVRAM on your Mac & Use safe mode to isolate issues with your Mac

As long as you have a good Time Machine backup your data should be just fine. If you are replacing the 2011 with a new Mac, before the new Mac is turned on for the first time, connect the 2011's Time Machine External Hard Disk to the new Mac using a USB cable. When you turn on the new Mac for the first time, it will launch Setup Assistant, SA will eventually get to a point where it asks if you want to migrate from another Mac, select Yes and then follow the on-screen prompts. It's really that easy and you should be up and running on the new Mac in no time.

Glad to hear you have been using Time Machine!!

Are you saying all these years you have had this computer and you never used Time Machine????