If you're dependent on the indulgence of relatives, I'd say you're pretty much out of luck. You should see just how much it will cost to fix your Xbox and whether that is much less than a used unit.
If it were your own money there are a number of things you can do to build up some credit at a major retail chain. Almost every chain has its own branded credit card that pays back in store credit. I have a Toys R Us Visa that was used for all of my monthly bills at a time when I had a lot of work expenditures that were repaid by my then employer. So most of my PS2, Xbox, and GameCube purchases at launch were covered by the store credits from the card.
A few other methods can be used to get gift cards for major chains but they're pretty slow if you don't make any purchases connected to the promotions. Back in the 90s there were a zillion of these sites and nearly all of them imploded but one legit site survives,
www.mypoints.com pays you read emails from various advertisers. You usually get 5 points for message you acknowledge. It's kind of like getting paid to receive spam but less annoying.
Currently it takes 3000 points for a $20 Toys R Us Gift card. I've gotten about $200 over the years but this is almost exclusively for just acknowledging the email offers and very rarely participating. If you actually sign up for something, buying a DVD from Fox Home Video, for instance, you get a big pile of points. Plus any shopping you do through their site pays off in more points. (Surveys can pay as much as 100 points but tend to ask ridiculously dumb questions that can really exceed my patience. Like, what qualites would I ascribe to this bank if it were a person. I'm not kidding.) Once in a great while these offers are actually decent and led me to indulge in a few box sets from Fox and some other stuff.
The payoffs used to be a lot better. You could earn points faster and the amounts needed for an item were lower. They used to give you 1,000 points just for getting another person signed up. Now it's just 100 points. Not that I'd turn it down, so if someone wants an invite just PM your email address.
So, depending on how much you mind clicking on a few emails daily, you can get something for nothing. Not a lot but every gift card adds up. I currently could muster about $150 in store credit at TRU but there isn't anything I currently want. I may break down and get a Xbox 360 as the oldest credit are on the verge of expiring. By then I should have about $200 in TRU credit and can get the 360 Premium for $200 plus games and accessories. It'll depend if there is anything I really like at a low price by then.