Fallout – More companions in Fallout: New Vegas

So, there's two seperate questions that could be answered here.

Can you fight unarmed, using primarily punches, and wear boxing gloves? Absolutely. In fact, Unarmed is among the stronger combat specializations in the game, though Boxing Gloves are somewhat weak specifically, because they cause only subdual damage - meaning they will knock your opponent over, but not kill them. Useful sometimes, but rarely.

Can you fight in a 'boxing match'? Well, not per se, however, there are two opportunities for you to engage in a sporting fight for some caps. The Thorn in westside is home to one such opportunity, where you'll have the opportunity to fight off various monsters of the Wasteland such as Radscorpions, Giant Mantises, Deathclaws and Cazadors. Any weapon is fair game in the Thorn, though you're welcome to go in unarmed if you're looking for a challenge.

Male (and only male, the legion are nothing if not chauvinistic) Players aligned with Caesar's Legion meanwhile, have the option to fight in The Arena, located at The Fort. In the Arena, the only weapons permitted are Machete's. All your other weapons are confiscated at the beginning of a match, and you are of course free to use your bare hands instead if you so desire.

Goodsprings has most of what you're looking for, I didn't use the school house, instead I used the boxes next to where you took target practice, both work benches are there. AFAIK you only need 1 box to store a lot of stuff. If it's because of sorting, well you can click on the top of your inventory window, click on the "Items" in the title bar, that will give you different types of sorts.