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Daniel,
to be able to host a shapefile/feature service, you would need ArcGIS Server.
You can however zip your shapefile and add it to a webmap.
Sign into your arcgis online account, go here http://www.arcgis.com/home/webmap/viewer.html?useExisting=1 , on the top left click on Add -> Add layer from File and navigate to your zipped shapefile.
Now save your map, this will produce the numbers you need in your navigation bar.
Hope this helps!
Tim
Daniel,
make sure you set your permissions of your map to share with everybody. On the upper right of your map click on "Share" and then check "Everyone (Public)". This should make it work.
Tim