Hi,
Does it indicate that the battery is charging when you connect a charging cord from it to a computer or an appropriate adapter?
If it doesn’t or even if it does try replacing the battery and then check what happens. Fully charge the replacement battery after installation, before attempting to turn on the headphones.
Here’s the ifixit [guide|133715] guide that shows how to do this.
Replacement batteries are available online. Just search for ''Sony SP73'' to find suppliers that suit you best.