Many people just leave the Lightning earbuds plugged into their iPhone and wind them around the phone, putting the whole assembly into their pocket. (I've damaged enough cables of expensive devices that I've developed a tic against doing that, but in the case of the iPhone earbuds, it seems to be exactly what they're designed to do, judging by the extra plasticizer in the cable shielding that's not necessary for comfortable wearing and makes them degrade faster under UV.)
> and Bluetooth headphones are still expensive.
Have you checked Amazon lately? There are (high-quality!) $30 wireless earbuds from reputable brands.