I had exactly the same problem. Interestingly though, I had only ever paired the Fairbuds with my Kindle, not with my Fairphone (FP3), and my Fairphone’s bluetooth was turned off. The Fairbuds would turn off every 5 minutes when listening to audiobooks on the Kindle. The phone was probably always in bluetooth range when this happened.
Installing the Fairbuds app and pairing the Fairbuds with the Fairphone fixed the problem. Now it works regardless of bluetooth being on or off, or the phone calls option being on or off.
Somehow the Fairphone 3 seems to have caused the Fairbuds to turn off every 5 minutes despite never being paired and despite bluetooth being turned off. I would have never suspected the phone as the culprit without reading this thread.