At 27″, it’s too big to “simply” take in to the Apple store. In desperation, I followed this support article from Apple that’s for older iMacs and officially no longer supported.
Which is a pity, because in typical Apple fashion, they’ve deprecated an article that’s still accurate and useful. Following the steps in that article (copy/pasted below in case Apple ever deletes the webpage – something they’ve a habit of doing, sadly), my fans roared to life and the iMac REVIVED!
Resetting the SMU can resolve some computer issues such as not starting up, not displaying video, sleep issues, fan noise issues, and so on. If your computer still exhibits these types of issues even after you’ve restarted the computer, try resetting the SMU. To reset the SMU on one of these iMacs:
Turn off the computer by choosing Shut Down from the Apple menu, or by holding the power button until the computer turns off.
[NOTE — in my case, OS X had crashed completely, so there was no “turning off” to do – I had to yank the power cable out, and after that, it wouldn’t switch on]
Unplug all cables from the computer, including the power cord.
Wait 10 seconds.
Plug in the power cord while simultaneously pressing and holding the power button on the back of the computer.
Let go of the power button.
Press the power button once more to start up your iMac.
I then had to repeat the process – but with a *fifteen second* wait, as per the support page that seems to supercede the article above – to get the fans to shut up.