ok when posting question you should provide more information. First what product your using, arcgis desktop, server, engine and which version? Secondly within that product what are you using? In desktop it could be ArcMap, ArcCatalog, Globe, Scene.. In Engine, which control? In server which API? Then which development environment (.net, java, python) and the type of customization (add-ins, extensions. etc.)
I am going to take a leap and assume it is desktop customization of ArcMap 10.x with .net and add-ins or extension code.
The ArcMap graphics layer is stored in the map document (mxd.) It is unique to this document and cannot be shared and used much. It can contain multiple geometry types (line, point, polygon, text, etc.) but no attribute information. As such it is not GIS data because it contains no attributes. GIS data is stored in a database of some sort, even if the database is a flat file. It is a geometry with geographical coordinates and a series of attributes that describe it in some what. In ArcGIS, GIS data is stored in a geodatabase or shapefile (usually.) In the esri world, a feature class is a set of geographic features that share the same type (line, point, polygon, etc.), the same coordinate system (geographic, lambert, etc.) and the same attributes and are stored together in the same file (usually a set of related files) or data table (usually a set of related tables.)
A layer in the map is a reference to a data source, the feature class and symbology, sub -queries and other information on how to draw the feature class on the map. A layer files (.lyr) is the layer in the map serialized to a file, it does not contain any features, only information on where the feature class is located and how to draw it.
I can't tell you what to do because I don't know your requirements, that is up to you to figure out. Saving geometries in graphic layers is not very useful. Saving them in .lyr files is impossible. You should look into saving them into feature classes in a geodatabase.