Depends on a few things:
1- Whether or not or Xbox is (still) covered by warranty. You can check this either contacting the place that sold you the Xbox, contacting Microsoft or simply checking the package for information.
2- If it is indeed covered by warranty, it depends on what kind of issue you had. If it is some kind of manufacture problem, some bug, or anything of that kind the guarantee PROBABLY covers it. If you have caused the damage yourself, the warranty PROBABLY won't cover it. Either way, you can contact them and ask for them to evaluate it.
If the damage is fully covered, it won't cost to get it fixed. Otherwise, it will.