Yeah an Apple Watch with LTE. Not sure if LTE is required, but needed for streaming. Could probably store data on watch before heading out. Need Bluetooth headphones. Battery life is ok.
LTE is not required. I use my non-cellular Apple Watch on runs and listen to either music or audiobooks. You sync your Apple Music music to the watch via the iPhone's Watch app. Audiobooks are similar, but since most of mine were from Audible I use their app to sync to the watch.
I have Garmin Fenix 6s pro and this variant has 32gb that you can use to store music. It's a nice watch, but the ergonomics of the buttons are not great compared to touch sensitive watches. It will do what you want though, and I love it for tracking my swimming laps!
Apple watch can save a few MP3s. My really old one can save at least a couple of albums and I never carry my phone with me. The watch connects to bluetooth headphones.