2011 iMac Hard Drive dead

Technically yes an external HD would work however it would be EXTREMELY slow. On a 2011, if the HD has expired your best bet would be to simply replace the whole computer. You could install an internal SSD however you would be left with a computer:

• worth less than $100

• Totally obsolete and only able to run High Sierra

Please use your time and money to replace it with a new Mac. You have MANY options available to you but the 2011 has seen it's best days (it is about 13 years old) and it would be best to replace it.

Last of the iMac’s with a internal CD/DVD drive.

Still a decent computer for its age…

A 500 gigabyte Samsung 870 EVO from NewEgg is $50. One terabyte is $70.

Make sure the seller is NewEgg..

You will need a tool kit, a 2 1/2 to 3 1/2 adapter, both made by Newer, and a temperature sensor, all available from Other World Computing.

https://eshop.macsales.com/item/NewerTech/TOOLKIT14/

https://eshop.macsales.com/item/NewerTech/ADPTADRV/

https://eshop.macsales.com/item/OWC/DIDIMACHDD11/

Assuming you still have the original 4 gigabytes of memory, I would snag two 8 gigabyte memory chips too…

Not hard to take apart, just take your time, and put the cat in another room…