This could be down to a number of things, including: Overheating, HDD Corruption, Problems signing into Xbox Account. In my experience, the latter happened to me.
Try disconnecting your internet, then turning on your Xbox. If you get to the dashboard, then chances are the signing in was the problem. To fix this try going:
- Xbox Dashboard > Settings
- Find the Automatic Sign-In feature and turn off
- Turn off Xbox, connect internet and restart the Xbox
- Try loading a game before signing in to your account.
Note: If it is not an Xbox Live account, you won't need to disconnect the internet.
If you didn't get to the Dashboard after disconnecting the internet then unfortunately something hardware wise has probably gone wrong.
Overheating could be the answer, in which a good clean out of dust and perhaps buying some extra fans may help. However as @Studoku said, this could void warranty if it wasn't bought second hand and that the console normally displays the "Red Ring of Death" when there is an overheat issue.
As for HDD corruption well there's not much you can do here apart from getting a new HDD I'm afraid. To check to see if this is the issue, insert a Xbox Memory Stick or a USB formatted for Xbox (if you have one) and remove the HDD, then try and restart the Xbox. If the Xbox loads properly, then there is something wrong with the HDD. (I'm assuming this is a normal Xbox 360, or an Elite version. If it is a Slim version, then I'm not sure if you can remove the HDD)
Hope this helps and it gets fixed. :)