Is an old Airport Extreme worth repairing?

If "quit working" means that the indicator light will not turn on at all......then the internal power supply has likely failed.

Repairs are possible.......but having done this, I would caution that they are not easy, since the AirPort was not designed to be taken apart. I would do this for fun, and not with the expectation that the device will start to function again.

A shop could probably do this, but the cost would likely exceed that of a new router.

A guess would be that you would want to try a new power supply.

PA-1600-9A Power Supply 60W For AirPort 2013 Extreme A1521 & Time Capsule A1470 | eBay

If this "five-year" old AirPort Extreme is an 802.11ac "tower" model, you may be better off replacing it vs. trying to fix it. Of course, we are assuming that you have already tried "factory default" resetting it ... even multiple times to try to "revive" it ... correct?

You have a choice of three types of resets, soft, hard, or factory default. I would suggest either of the latter two, and again, you may need to try them more than once.

Ref: How to reset your AirPort base station - Apple Support

It is important to get the model correct.. Tesserax asked if it is the tower AC one.. or previous model flat one..

The actual fix is different depending on the model.

And I agree with Bob.. this is a DIYer bit of fun.. 5 years is already excellent for domestic type router. It is time for a replacement anyway.