You're not alone
The issue you are having is that the shapefile you've created from the OSM data doesn't contain any closed polygons as defined by the shapefile standards. The data is a series of polylines that may (or may not) be an enclosed building or a walkway between parks.
If you want the Chelsea section of New York City, you might want to look at this model in the 3D Warehouse:
If you want to import OSM data, I would suggest downloading and installing my OSM Importer:
Browse to http://www.openstreetmap.org/ and use the native OSM exporter to grab an area:
Adjust the highlighted region:
The filename defaults to "map.osm" and I changed it to "denmark.osm". Note that even a relatively small region can result in very large files.
Next run the OSM Importer and select the file:
Wait a bit ... you should eventually get a popup (depending on the size of the file):
I'm using inches so I scaled it up by 1,000 ... depending on the geometry, you may need to increase this (or decrease it):
If all is well and the planets are aligned correctly, you should get something like this as your reward: