The chart object currently does not have an option of being exported or saved separately. Such a functionality was requested for calc here - https://bugs.freedesktop.org/show_bug.cgi?id=30944.
Proposing to add this functionalities to writer (and later port to calc as well) :
1) Save the object to a image file
2) Export the object to a PDF file.
Only for the chart object.
Actually the same request than 30944
*** This bug has been marked as a duplicate of bug 30944 ***
This is a Writer issue, therefore changed the 'Component' field accordingly.
This is an enhancement request, therefore changed 'Importance' accordingly.