Is it possible to extract a network feature dataset from ArcGIS StreetMap Premium to an enterprise geodatabase?

No. It is not possible to extract a network feature dataset from ArcGIS StreetMap Premium to an enterprise geodatabase.

The StreetMap Premium dataset contains a detailed basemap data, locators for geocoding, and a network dataset for routing (Routing_ND). The network dataset provided in StreetMap Premium is set to read-only. Refer to ArcGIS StreetMap Premium: Technical Restrictions for more information.

Note: 
Esri recommends creating a network dataset in ArcGIS Desktop, and storing it in an enterprise geodatabase. To create a network dataset in ArcMap or ArcGIS Pro, refer to ArcMap: Creating a network dataset or ArcGIS Pro: Create a network dataset respectively.