Sounds like there is some kind of issue with your server. Unless it’s a dedicated, try to contact your host. They should be able to provide some advice. My first guess is that the downloader script is timing out. You can try to download the full package instead, it is usually a more reliable way to install Magento.
There are two main reasons why you may get this error.
1. There is an actual problem with the server, resulting in the 503 error response. In this case, you should contact your hosting provider.
2. Your store has been put in maintenance mode. If Magento sees a file named maintenance.flag in the root directory, it will automatically redirect all requests to the default 503 error page. This can be used during upgrades, etc. to prevent users from accessing the store. The solution is, as already mentioned, to delete/rename the maintenance.flag file.
There might also be a good idea to create a more user friendly “maintenance page” than the 503 error, which is kind of misleading for customers.
I had same error when installing a new theme via Connect Manager, the install never completed and I had to delete the maintenance.flag file before I could get back into the system.
