In the name of Allah, the most Beneficent, the most Merciful.

Answer

Before answering your question first of all we will look at the harms and severities of the sin of adultery and fornication in Islam.

1. The Prophet of Allah Sallallahu Alahi Wasalam has said that whomsoever commits illegal sexual intercourse or consumes alcohol Allah shall extract from him the Iman (faith) just as how a person removes his clothes from his head. (Kanzul Ummal)
2. The Prophet of Allah Sallallahu Alahi Wasalam has said, “Abstain from Zina for indeed it has four traits, it eradicates beauty from the face, it restricts sustenance, it angers Allah and entry into hell. (Kanzul Ummal)
3. The Prophet of Allah Sallallahu Alahi Wasalam has said that Allah’s anger becomes intense upon the fornicators. (Kanzul Ummal)

Islam being a divine religion has given us rules which act as precautions to stop someone falling into this heinous crime. Therefore Islam has strictly prohibited gender interaction except where there is a need.

In regards to your question it is not permissible for women to go to a mixed gym where the rules of hijab and concealing oneself are not taken into consideration and where there is free intermingling of the sexes.

Only Allah Knows Best.

Mohammed Tosir Miah

Darul Ifta Birmingham