What is the ruling on going to a gym for weightlifting where music is constantly present, given the unavailability of another place to practice this sport, which is essential for strengthening the body?

The default ruling for games is permissibility. However, it is forbidden to listen to musical instrument sounds, due to the hadith (definition): "I was forbidden from two foolish, wicked sounds: a sound at the tune of a devil's flute, and a sound at a calamity with a devil's wailing."

If you cannot change the reprehensible act or hope for the music to be stopped, then it is not permissible for you to be present. This is because the end does not justify the means, and the necessity that permits the forbidden has conditions, among them that the necessity reaches a level of unusual hardship, that it is specific (meaning no other way exists), that it is limited to the extent of the necessity, and that it does not contradict a religious text or a stronger sound analogy.

If your denouncement removes the reprehensible act, there is no harm in going. Otherwise, you must distance yourself from the place of the reprehensible act, due to the Almighty's saying: "And He has already revealed to you in the Book that when you hear the verses of Allah being denied and ridiculed, do not sit with them until they engage in another conversation. Indeed, in that case, you would be like them." So, whoever sits in a gathering of disobedience by choice is like them in sin.