Normally you cannot run Microsoft's asp.net AJAX on the 1and1 servers because they have not installed it, and you cannot get it to run in a partially trusted environment.
But do not despair, older versions of Microsoft's asp.net AJAX, code named ATLAS, an still be downloaded from Microsoft.com and ATLAS does not need to be installed on the server. You only need to copy the Microsoft.Web.Atlas.dll to your bin directory.
There were two releases of ATLAS, the April CTP (community technology preview) and the June CTP. You need to download the April CTP here since it seems the June CTP needs installation.
Here is my example page on 1and1. My example uses the UpdatePanel control.