It seems to me that your best bet that will combine portability and modernity is an mp3 player.
Despite the rise of the smartphone and its effects on dedicated mp3 players — the iPod peaked around 2005/6 and steadily declined till 2014 when sales numbers were last available — similar devices continue to exist under a plethora of brand names.
Many are marketed with your use case in mind.
For example, this one (picked after a very cursory read of a couple reviews) is lighter than most phones and claims to be safe for sports through shock absorption.
It also resolves the cable issue through Bluetooth connectivity for earbuds.
A further step in that direction is the clip form factor for mp3 players, e.g. this one .
This has no base unit — the earbuds with their wraparound band are themselves an mp3 player.
What a time to be alive.
I can't think of a better way to get portable music without bringing a serious device.
That said, I also like Willeke's suggestion.
As someone who plays and writes for piano, I do find that my attention can sometimes be fully occupied by mentally playing or coming up with music.
Can't hurt to try before buying a device.
lifehacks.stackexchange.com/users/22214/Luke Sawczak
I'd recommend cordless earphones with a radio antenna.
If you want your own music then you would have to get cordless earphones with some storage.
lifehacks.stackexchange.com/users/27565/AdrienDaBoss
I've bought "Mighty" player for my daughter a couple of months ago.
It is quite small and claims to repel water.
But this is meant to be used with Spotify so not sure if it fits your purpose.
It has bluetooth
so wireless headphones could be used with it although I have not tried this feature yet as use earbuds from older iPhones with normal audio jack.
On the negative side I think it is too pricy for such a device you have to constantly sync with Spotify and apply updates via their app.
It is cumbersome at times and sometimes you leave it at home just because there is no time to sync it before going out.
lifehacks.stackexchange.com/users/27583/Aivaras
What I do is, keep my phone on air plane mode so I can listen to songs saved.
It is as good as keeping your phone at home(no notifications).
Or if you need to connect to internet for Spotify, you can choose do not disturb mode if that helps.
Last option is as mostly suggested with a dedicated MP3 player / ipod
lifehacks.stackexchange.com/users/26946/Vajra
There are headphones that include a small memory.
It's only a simple cable that goes behind your neck.
You can download some music there and won't be bothered with lots of cables or having to carry multiple devices: https://www.amazon.es/Sunstech-Triton-Reproductor-negro-naranja/dp/B005DKJ40O/ref=sr_1_2?__mk_es_ES=%C3%85M%C3%85%C5%BD%C3%95%C3%91&crid=RN1TQUP1EG17&keywords=auriculares+con+memoria+interna&qid=1557233656&s=gateway&sprefix=auriculares+con+me%2Caps%2C154&sr=8-2
lifehacks.stackexchange.com/users/18454/Mario Garcia
So currently there is a lot of runner watches that can store and play music using bluetooth headhphones.
I know at least two such watches.
Garmin Vivoactive Music and TomTom Runner Cardio+Music .
These running watches are designed to be much less distractive than mobile phones or smartwatches.
The additional advantage of such watches is that you can easily keep track of your pace without connection to your phone
so you don't have to take phone with you.
Of course you will have to choose a good bluetooth headphones but currently there is plenty of them designed for runners.
lifehacks.stackexchange.com/users/9461/running.t
I love to practice exercises, but when going out for a run, I never take any electronic devices  (by electronic devices, I mean any devices that may distract me.)  in order to stay focus.  But I wish I could still listen to music, how can I achieve that without taking any phone or smartphone with me ?   Also I don't want to be annoyed by cables that could possibly dangle.
How can I listen to music while running, but without smartphone?
lifehacks.stackexchange.com
<sports><music>
lifehacks.stackexchange.com/questions/20704
lifehacks.stackexchange.com/users/26603/Ced
There are many ways to listen to music while running without a smartphone. One solutions would be to get an mp3 player with some bluetooth earphones. There are also running watches available that provide this functionality. Alternatively, you could buy some earphones that have a radio antenna or some built-in storage. The options are to use an MP3 player, antenna earphones, a Mighty player, your phone on airplane mode, headphones with memory for music or a runner watch.
There are many ways to listen to music while running without a smartphone. One solutions would be to get an mp3 player with some bluetooth earphones. There are also running watches available that provide this functionality. Alternatively, you could buy some earphones that have a radio antenna or some built-in storage. An mp3 player or ipod seems like your best bet or a set of headphones that include a small memory. You can also get runner watches that can store and play music using bluetooth headhphones.  
A dedicated mp3 player, cordless headphones with and antenna or memory to store music, or watches for runners that store and play music using Bluetooth headphones may be suitable for your needs. The simplest solution is probably an mp3 player or iPod. However, other potential options include headphones with built-in storage or a radio antenna or a sports watch that includes music storage.
Try an MP3 player. Try erphones with radio antenna. Try a Mighty player. Use your phone on airplane mode. Try headphones with memory for music. Use a runner watch.
An mp3 player or ipod seems like your best bet. There are headphones that include a small memory. At the moment there are a lot of runner watches that can store and play music using bluetooth headhphones.  
The simplest option is probably an mp3 player or iPod. There are headphones available that have either music storage or a radio antenna. There are runner watches available that can store music and connect to headphones.