Is there a way to export/save the settings for a network dataset? Setting up the connectivity groups, attributes, evaluators etc can take quite a while, even for a small dataset. If the network dataset needs to be recreated multiple times for whatever reason, it quickly becomes a painful task.
No, you cannot do that with the core tools of ArcGIS. I guess Esri is working on that and this (I hope) should become available in some next release (at least via Python - arcpy.na module).
Meanwhile, you might consider taking advantage of the Street Data Processing toolbox code to make it applicable for your use. It's been written in C# and the source code is available.
There are also two ideas at ArcGIS Ideas portal you might vote to promote this idea: http://ideas.arcgis.com/ideaView?id=08730000000bqE6AAI and http://ideas.arcgis.com/ideaView?id=087300000008DxlAAE.