What version of Fuel are your using? What version of PHP does 1&1 run? And what type of webserver?
Are you sure all your file rights are set ok?
You get this error if an Exception occurs in PHP's shutdown event. FuelPHP uses this to clean up and to write session information. Often, it is related to that (error in writing sessions).
This is always a challenge to find, since PHP is already is shutdown mode, and the exception handler isn't operational anymore, so PHP can not give a proper error message.
5.3.5. is an antique, but it should be able to run Fuel without problems.
I'm afraid you'll have to start debugging.
In ./fuel/core/bootstrap.php you will find the "register_shutdown_function". Add some logging to the different steps, and see how far it gets.
I assume you have checked the logs (both the webserver log and the fuel logs for any error messages? Have you enabled all logging in your config.php (L_ALL or L_DEBUG level)?