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 do not have a file maintenance.flag in the root directory, but the error \"Service Temporarily Unavailable
The server is temporarily unable to service your request due to maintenance downtime or capacity problems. Please try again later.\” still there.