Mac OSX 10.6.8 QGIS 1.8.0 : Using the field calculator from the attribute table, the $area of each feature is shown nicely in the table until I try to save it, then all the values are replaced by NULL so are no longer available for use. Furthermore, if the column is left there and the field calculator asked to update it, all values remain NULL. Any fix?
Two chances to get it right:
Check if the Layer CRS has metres as units, not degrees
Expand the width of the field. Square metres might easily exceed the limit you set. The shapefile database format is rather strict in that respect.
I've experienced the same with 1.9 dev version when creating a new column and immediately writing values into it without stopping editing mode. However, when I stopped editing and turned it on again, it worked fine.
Many years and versions later I had a similar issue. In my case the issue was that my file was saved as a KML. I exported to a shapefile and all was resolved.