If your AirPort Extreme is flat shaped, and looks like a small white pizza box with rounded corners, there is a good chance that the outboard power supply "brick" is failing. There's still enough power to run part of the AirPort....like the Ethernet ports....but not enough to provide enough current to allow the AirPort to power up and operate normally.
If you are willing to gamble $25 that a replacement power supply might bring a bit more life to the AirPort, you can order from the source below, which I have used in the past.
https://www.savingology.com/ProductDetails.asp?ProductCode=AIRPORT-EXTREME-MA073LL-A&click=2&gclid=Cj0KCQiAnNacBhDvARIsABnDa69ssDWBQsTWSkygZY2hI2zsSUEu2oNO_yJecmzsTAvwOgbsMiZffHYaAiGiEALw_wcB
There may be other sources as well that might allow you to return the product for credit if it does not do the job. Google "AirPort Extreme Power Supply"
If your AirPort Extreme is shaped like an 8" tall tower, things are a lot more difficult because the product was simply not designed for service. Repairs.....if parts can be found.....would cost far more than the original cost of the AirPort Extreme.