iMac 27-inch late 2013 Computer Won't Start Up

What you’re seeing points very strongly to a failed or failing internal hard drive. The flashing folder with a question mark means the Mac can’t find a valid system to boot from. The globe with an exclamation mark and error -4500F confirms that Internet Recovery is also failing—usually due to either network compatibility issues or a disk that can’t be initialized at all. This behavior is extremely common on this generation of iMacs with original mechanical drives.

First, I pretty much agree with the others that have replied to you. However, there are a few things you can try (some have already been mentioned) to either revive it, or at least, confirm the diagnosis. The goal is to determine whether the logic board is healthy and whether the drive is recoverable.

• Reset NVRAM. Ref: Reset NVRAM on your Mac - Apple Support
• Try Internet Recovery on a wired network
  • Plug directly into your router via Ethernet
  • Power on and hold Option + Command + R
  • If the globe appears again with -4500F, move on
• The next step would be to boot to Recovery mode, and then, use the Disk Utility to check the Mac's internal drive, but since you were unsuccessful in accessing this mode, you can just skip this step ... however, it still may be worth a try.
• Test with an external boot drive. (This was mentioned by others). If the iMac boots successfully from the external drive, the internal drive is 100% the problem.
  • Ref: How to use an external storage device as a Mac startup disk - Apple Support

A suggestion on how you may still be able to use this Mac until you can get a replacement:

If the external boot works, you can run the iMac entirely from an external SSD. This avoids opening the iMac and is fast, stable, and inexpensive.

Note: If you need data from the internal drive and it still shows up intermittently, recovery software might help—but if it’s invisible in Disk Utility, it’s beyond DIY recovery. So, if any of that data is critical to you, you will have to solicit help from a professional recovery service (unfortunately, at a cost close to a new computer).

Samwise-100 wrote:

While I used Time Machine successfully for a number of years I had trouble running it a few years ago, couldn't resolve the issue and, unfortunately, haven't run a proper backup since then. I won't be in a position to replace the computer till the middle of the year, and I need to get it up and going again ASAP.

If your computer's internal drive is corrupted – or failing, which could well be likely on a 2013 machine – and you have not run a proper backup for several years, you are up the proverbial creek without a paddle.

Time Machine backups are not bootable, so you will need to get a working operating system onto that machine, somehow, before you can restore your old Time Machine backup. That may require use of another old Mac – by the time your Mac was released, Apple may not have been shipping compatible versions of the operating system on physical media any more. (I don't know if even an original – and hard to find – Apple USB flash drive copy of Mountain Lion would work with your Mac, as your machine requires Mac OS X 10.8.4 or later. And all later OSes were only distributed electronically or preinstalled on Macs.)

If you have a bootable clone backup, you could try booting and running from it.

Please follow the steps in If your Mac doesn't turn on - Apple Support

Remember it is now 13. years old and totally out of date. Consider replacing it, you can use your Time Machine backup to restore your data, apps, settings to the new Mac.

There's one thing you can do to see if it's the boot drive that's failing or something else: boot into Target Disk Mode. If you're successful you'll get a screen link this:

If you're not successful it's pretty sure the drive that's the problem.

Samwise-100 wrote:

While I used Time Machine successfully for a number of years I had trouble running it a few years ago, couldn't resolve the issue and, unfortunately, haven't run a proper backup since then. I won't be in a position to replace the computer till the middle of the year, and I need to get it up and going again ASAP.

You need to take a step back and realize that in all likelihood, your data may very well be gone for good.

You also need to process the fact that your Mac is obsolete. Apple no longer offers repair services or parts for that machine.

I understand you need something, but trying to get this machine running again is probably an exercise in futility. Even if you manage to get it to boot successfully and get macOS re-installed on either the internal drive or an external drive, your data is probably gone.

Don't throw good money after bad.