Being mentally worn out just kinda makes you feel like shit. It's a terrible state to be in, you don't want to do anything, but doing nothing also feels bad.
A lot of our entertainments and hobbies, especially in urban areas, are mental to begin with. The most evident example would be scrolling on your phone, which evidently is 100% a mental thing, but there's a lot in this box, videogames, music, reading can all really demand a lot from you mentally.
Explicitly physical things are not so common. Gym. Dancing. Climbing. Running. All of these are things that people who work with their brains often talk about as "relaxing" or "great against stress", even if they can put the body until great physical strain.
It is my experience that when mentally worn out, the "don't want to do anything" really translates to "don't want to do anything mentally", problem being, thinking about doing the physical thing instead is actually a little bit of mental, albeit more emotional, work, but once you ACTUALLY do that, then after spending the whole day focused on deep, complex stuff, being the caveman bench pressing is actually absolute bliss.