Hi Ben
Sorry about the issue preventing web maps from opening in ArcGIS Desktop 10.0. The fix for this issue (NIM065725) is in the upcoming 10.0 Service Pack 3. This will allow you to open ArcGIS web maps in ArcGIS Desktop 10.0. When you open a web map in 10.0 Service Pack 3, it gets converted into an MXD file.
Opening ArcGIS web maps isn't supported in 9.3.1, but if there is a particular map service used as a layer in a web map that you would like to use in any version of ArcGIS Desktop, you can make a connection in ArcCatalog to the server from which the service is being served. Look in the details page in the ArcGIS.com entry for the web map to see a list of the services it contains.
Note: The support for web maps in 10.0 Service Pack 3 doesn't provide support for all the different types of data found in a web map. It will support ArcGIS map services, ArcGIS image services and OGC WMS services in web maps, but not web map layers based on uploaded shapefiles, CSV files, GPX files, KML network links, or note features (features simply drawn on the map). Support for this wider range of web map content is planned for the ArcGIS Desktop 10.1 release in Q1 2012. Sorry again about the lack of full compatibility at this point in the desktop clients with the latest functionality in the ArcGIS web map format.