Hi - There's no way to do this from Explorer Online; however a colleague has just sent me some very useful info that will hopefully help - you should be able to achieve what you want in the upcoming versions of ArcGIS Explorer Desktop, and ArcGIS (10.1) for Desktop.
In the next ArcGIS Explorer Desktop release you'll be able to make a layer package from your notes, which could then be loaded into ArcMap (from where it could be saved out as a shapefile). In the current ArcGIS Desktop, feature collections are ignored if you open a web map, but in the upcoming ArcGIS 10.1 for Desktop release, when you open a web map the feature collection layers will be automatically converted to feature class layers, which can then be exported as shapefiles. Hopefully one of these solutions will help.