Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.
There is no harm according to Islamic Shariah for a woman to do what will strengthen her mind and body, like physical and mental exercise.
However, this should be according to requirements of Islamic jurisprudence, like being dressed properly, not mixing with men, and not to wasting time, and performing prayers and other obligations.
A married woman should not go out without the permission of her husband.
So there is no harm for her to learn what will protect her body, her honour and protect her religion from cultural and ideological invasions.
The Prophet (Sallallahu Alaihi wa Sallam) said: "A strong believer is better and more lovable to Allah than a weak believer, and there is good in everyone, both the weak and the strong believers but cherish that which gives you benefit and seek help from Allah and do not lose heart." [Reported by Muslim]
The questioner stated that it is not permissible for a woman to take her clothes off outside her home. In this regard there is a Hadith which is reported by Abu Dawud that the Prophet (Sallallahu Alaihi wa Sallam) said: "When a woman takes off her clothes in a place other than her home, she tears the veil between her and Allah, The Exalted". Al –
A woman is ordered to cover herself lest an alien man would see her.
Some of the people of knowledge said that the Hadith applies to a place where she could be seen by an alien man. However, if the woman is in a safe and secure place or alone, then it is desirable for her to cover the Awrah. This is the opinion of the Maliki school of thought and those who agree with their opinion.
In Minah Al – Jaleel Sharh Mukhtasar Khaleel, which is a Maliki book, it is stated: "It is desirable for a man or a woman who is not praying to cover their Awrah, meaning their private parts and the parts near the Awrah be it of a man or woman, and it is disliked to reveal the private parts for no reason when being alone. But it is permissible to reveal what is beyond Awrah when being alone. This is what is relied upon."
Therefore, it is permissible for a woman to practice sports if this is in a safe and sealed area which is far from the access of men, but she has to cover the Awrah from other woman, and if she covers the whole body, then that is better
Allah knows best.