late 2013 imac hard drive failure

They may. This link to Apple Repair services should get you started.

Apple Repair and Repair Status Check

Some additional things to consider. Some iMacs can have their HD's replaced and some can not. (Same for RAM.) I had a 2013 iMac which could not, so I decided to replace it. You can use the "About This Mac" under the Apple menu to get your Mac's model and serial number. Then the Apple support site has a link which will tell you if your unit is able to be upgraded or not. If you can not replace your unit, consider attaching an external drive with a current OS and run off of it.

Incidentally, today's drives are so large, you can also first partition it into two partitions and use the second (larger) partition for backing up your files. I personally do that and I like to keep my external OS separate, although that is certainly not necessary. Good luck.

A Late 2013 imac 27" hard drive is replaceable. I suggest you get an SSD, it will make a dramatic improvement in speed.

Using the same drive as your system drive to backup isn't a good backup solution. When the hard drive fails, you are left without a backup.

A bit like fearing the handle on your basket might fray, so you double the eggs in the basket! :)