The Secret Club's Secret
This sidequest is unlocked by examining the door to Gerudo's local secret club. But to save you having to guess a password and as a result waiting a whole in-game day before you can try again, let's snoop around and find the password first.
First, talk to the two Gerudo ladies gossipping at the local bar (the same bar that also makes the Noble Pursuit for the Shrine Quest "The Perfect Drink", by the way). They will see you trying to eavesdrop and of course will tell you to go away so that they can gossip in peace.
To hear what they are saying, exit the bar and go into the room behind them. Walk up to underneath the window to the right and you will see a prompt where you can press [A] to "Listen". Now you can hear them gossip about some sort of secret club, and they will even tell you the password.
The password is GSC followed by a diamond symbol. Remember this and go to the door now. It is on the left side of the alley to the left of the armor shop. Examine it and a voice will ask you for the password. Enter in "GSC" followed by the diamond symbol.
The Gerudo Secret Club sells exclusive armor you cannot find anywhere else, and this includes a special heat-resistant armor set designed for men. Even better, unlike the women's Gerudo Armor, you can take it to Great Fairy Fountains to be upgraded. You can also wear non-Gerudo Armor in the secret club and you will not be kicked out of Gerudo Town.
You can also get the Radiant Armor set, shown in the above screenshot, which makes you look like a glowy skeleton. In addition to the Rupees you will need to provide Luminous Stones.