It's a glitch which allows you to -- under certain circumstances -- purchase weapons or armor from a shopkeeper, then sell them back for more than you paid for them, and to continuously loop that process, thereby giving you anything you want from the shop, as well as all the shopkeeper's caps, essentially for free.
This YouTube video demonstrates it on PC, and I've used it successfully in both Fallout 3 and Fallout New Vegas (already!). By the end of my last Fallout 3 run, I had over six figures in caps. Also, for a guy with less than an hour into New Vegas, I think I'm doing OK financially -- almost 800 caps.
The trick to it is having in your inventory a weapon or piece of clothing/armor which the shopkeeper sells, preferably at a lower repair value. Sell it to him, and commit the transaction. Buy it back and commit. Lather, rinse, repeat. It may take two or three repeats before you start buying it at its original value but selling it as 100% repaired.
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.
Best Answer
According to the Fallout New Vegas Wiki removing the FNV patch can solve this problem. Instructions on how to do that are here.
After finishing that, you can reapply the newest patch.