iMac not booting - turning off at Apple logo

Hi everyone,

Hoping you can help here. My late 2012 iMac 21.5" with the 1TB HDD has stopped working recently. I initially suspected a HDD failure but now I'm not so sure. It boots up, plays the usual chime, the Apple logo and loading bar appear and then after 10-20 seconds it just switches off. Here's what's happened to date/what I've observed:

• The iMac had been in storage in it's original box in the cupboard for a 2-3 years, as I have a newer MacBook Pro. I've taken it out recently to access some old programs and photos. It was very well protected from bumps and knocks, but did become very dusty when I took it out of storage and there was dust in the vents.
• For a little while before this happened, I had a few minor incidents with it that I didn't take notice of at the time. When it went to sleep, there were a few instances where I had trouble waking it back up or when it did wake up, the system was suspended/frozen with either no response to inputs or a spinning wheel and a blurred out screen.
• I've run a disk utility repair/check on it and it has come back saying no problems.
• Here is where it gets weird - in disk verifier, I can see all of my files and navigate through the folders, so if the HDD is failing, it hasn't completely failed.
• My Windows partition will also not boot, I get a MS DOS-style error message when I try to boot into it
• I've tried re-installing Mac OS from both a recovery USB stick and the recovery partition (alt-D); neither work

Any ideas? At a long shot I was thinking perhaps it is a temperature issue, but I guess it's more likely to be a HDD issue of some sort? If the latter is the case, are there any bright ideas as to how I can get my files off the drive whilst it is still clearly exhibiting some form of life? The iMac dying isn't the end of the world to me (although I'd love to save it if I can!) as I have a newer machine, but I'd love to get the files off of it.