Mid-2011 21.5 iMac starts with gray/pink stripes onscreen

Hi, OFBG1792.....that indicates probably a video card failure/video card dying/ ribbon cable might have gone bad inside/isn't seated correctly, and assuming your iMac's built in LED screen panel is OK, that may be it. You could run Apple Diagnostics, reboot your 2011 iMac, holding down the "D" key on the keyboard, and be patient as it's running a number of tests in the background.... hold down the "D" key until you see some kind of diagnostic screen, eg, component on one side, and pass/fail/red "X" or green checkmark on the other side.... that should help you out, sort of... your only option would be to take it in for evaluation and replacement of the video card which isn't really worth it, financially.... I hope that you have done a backup of your iMac and are doing backups on a regular basis..... it may be time for you to start looking for a good quality and newer "Preowned" iMac or iMac pro... you may be able to, depending on your computing needs, and availability of used macs where you live,"get by" with an Intel Mac Mini, usb keyboard and mouse and standalone monitor. some of them have a built in optical drive, and some don't.

hope that helps you a bit

John B

+1 on a likely GPU failure. Due to how obsolete (likely at least 5-6 years) the computer is, it's telling you its time to replace it.

Assuming you have been backing it up using Time Machine then your data is safe. When you buy the replacement Mac use Setup Assistant and Migration Assistant to migrate. If you have not backed up and the computer will not run then you will need to open the computer and remove the HD, then place it in an enclosure such as a https://eshop.macsales.com/item/OWC/ME3NH7T00/ and then use the SA&M link I provided.

That is good news.