Sadly commands don't have square radiuses. However, you can use a spherical radius -- but that can have some complications. I would recommend using a scoreboard to track whether or not a player is in the arena. You could give them a score when they press a button to join the game or however you want to do it.
For detecting how many players are left, you should create a dummy scoreboard, and run the command /execute as @a run scoreboard players add players <scoreboard objective>. This will give you a count of how many players are left. Change the @a to have some target selectors to specify which players you want to count. Here's an example for a spherical radius: @a[distance=..100]
I'm still a bit new to command blocks, so I'm unsure on how to hook this up. Are they all impulse and unconditional? I'd probably know If I was better at command block programming, but I'm still learning.
If you don't mind, a picture would really help me to visualize it. Also, does this detect when there is one player left in the radius?