The Legion Arena is right below Caesar's tent and is comprised of a small pit with walls of scrap metal hastily put together to form a post-apocalyptic Colosseum. Here the Courier, if male, can fight slaves to the death for the amusement of Caesar's legionaries (females are not allowed to voluntarily fight in the arena). Each fight will pit the Courier against progressively tougher opponents. If Benny escaped from The Tops at the conclusion of Ring-a-Ding-Ding! he will be a prisoner at The Fort, having been captured by Caesar's Legion, and the Courier is given the option to fight him in Arena.
Speak to Otho, who is standing to the right of the arena gate, in order to participate in arena matches. When you enter a contest your weapons and equipment will be confiscated, replaced with Gladiator armor and a machete. All arena opponents will be either unarmed or equipped with machetes by default. Battling in the arena will give you fame with Caesar's Legion.
Because the Arena fights are as much for earning respect as they are for entertainment, Otho will not allow a female Courier to participate and will say "Know your place woman... only men are allowed to fight in the arena,". The presence of a female slave (Ranger Stella) in the Arena is not a contradiction to this policy as she is not there to earn respect, she is there to die.
Round 1: A fight against two slaves with machetes.
When you fight in the Arena, all non-quest items (even keys) will be removed from your inventory and deposited in the Equipment Storage case outside; after you win, each item in the case is returned to you with its own popup message (just like with Mojave Express dropboxes). To avoid minutes of popups, you can dump all your stuff in a footlocker found in any nearby tent. Assuming you don't spend too much time in the Arena, it should all still be there when you're done.
Note that even though the footlockers are marked as owned, retrieving your items from them does not count as stealing.
If you turn around quickly as the fight starts, you can open the footlocker and retrieve all of your belongings.
Since quest items can't be removed from your inventory, Motor-Runner's helmet and the rebreather can both be worn during your fights for a little extra damage protection and, in the case of the helmet, a handy +5 Melee boost.
ED-E, if it has a clear shot at the arena (e.g. at Caesar's tent), will attack the enemy combatants.
Rex may sometimes be transported into the Arena alongside the player. This has no negative consequences and makes the fights ridiculously easy due to Rex's high damage output.
After winning in the arena legion guards around the fort will remark on the Courier's bouts with the slaves and Ranger Stella claiming that they didn't think anyone could defeat a ranger in one on one combat.