Is it permissible for us to exercise in the mosque while we are menstruating, given that we were told it is not permissible to enter the prayer hall?

A menstruating woman is not permitted to enter the mosque itself, but she is allowed to enter its courtyards. As for a non-menstruating woman, she is permitted to enter it and remain inside.

Regarding exercising in the mosque: If another place is available, it is not appropriate, especially movements that contradict tranquility; because mosques were built for worship.

However, if no other place free from religious impediments is available, there is no objection to exercising in it. This is supported by the hadith of Aisha, may Allah be pleased with her, concerning the Abyssinians playing in the mosque in the presence of the Prophet, peace and blessings be upon him. Their playing was a form of training and preparation for the enemy, and the permissibility of this entails a religious benefit.