Can a 13 year old IMac be repaired?

The short answer is yes. Anything can be repaired. Before you arrive at the conclusion a repair is necessary read If your Mac starts up to an Apple logo or progress bar - Apple Support. Follow those instructions.

Sounds like the hard drive may be failing. For such an old computer booting from an external USB drive won't really be useful due to the slow speed of the USB2 ports. While it is not easy to open up an iMac, it is possible to replace the internal drive with an SSD upgrade. OWC has upgrade kits and video instructions. Keep in mind the videos make it seem much easier than it really is....the iMac has some very difficult to see & reach cables & connectors which are very easily damaged. Plus it is easy to damage the large & heavy LCD Panel...I have seen people crack them since they must be kept upright or they may crack under their own weight.

You should really first confirm the hard drive is bad. You can try running the Apple Diagnostics to see if any hardware issues are detected. If the diagnostics report a bad hard drive, then you have your answer. Unfortunately the diagnostics don't detect most drive failures. I have posted instructions for another user to use a bootable Knoppix USB stick to check the health of the internal hard drive...here is a link to that post with the instructions:

https://discussions.apple.com/thread/255081523?answerId=259452910022#259452910022

As others have mentioned, is it economical to repair or upgrade it? May depend on how you use it and your expectations...just remember it is risky.

That model would be obsolete and no longer repairable by Apple. You may be able to have it repaired at an authorized service provider, depending on the issue, but to be honest, it’s not worth it. The cost of repair would be more than it’s worth in working condition.

+1, A 13 year old computer really should be replaced! If you cannot afford to buy a new Mac then you have many options available to you, for example Apple's own refurbished machines. There is no downside to an Apple Refurb, they are identical to new, have the same warranty as new, are eligible for AppleCare and can save you hundreds of dollars. You can find Apple refurb's at www.apple.com and scroll to the bottom of the page and click the Refurbished section.

Another option is buying a high quality used machine from a reputable vendor such as Macsales.com. NEVER EVER buy from E-Bay, Craigslist, Facebook Marketplace etc. More often than not, these machines are not maintained well, are not setup properly for resale and in many cases you are simply inheriting another persons problems.