Derik,
Do you experience memory lack during the export of SDC dataset to a shapefile? In this case, I'd suggest exporting to a feature class in a geodatabase directly. Shapefiles are not the best storage format performance-wise 🙂
I guess you might get slightly better performance and control over the export process if you will use the Feature class to Feature class tool (from the Conversion toolbox) rather than selecting features in the Data View in ArcMap and then exporting. Thus, you could save the resources because you do not need to draw features on the map. In the Feature class to Feature class tool dialog box, you might use the STATE field (I believe it's been there) in the Field Expression field to export only those streets that have this attribute value (Washington).
Please do not use personal geodatabase for storing your data! They have 2GB per database size limit and often go extremely slow after 500MB. Use file geodatabase instead.
Let the software do the export over the night, for instance, since it can take some hours for sure and you do not want to sit and wait for the software to finish. You might see that ArcCatalog or ArcMap can go to the "Non Responding" state during the heavy load (like exporting data or building a network). Don't worry about it - let it do its job. A good way to check that the software has not been hanged is to use the Task Manager to see if the memory use for the application is being changed. If it does, then you are good. If it does not, wait for some time just to make sure it does not indeed and then restart the software and try again.
Maybe you could talk to your system administrator to see if there are any better machines at your disposal? Do you have x64 bits operating system? I do and should say it makes a huge difference when processing data (not just in ArcGIS). Remember that WinXP x86 has 4GB RAM limit (128GB for x64); Windows 7 x86 has 4GB limit, too.
If nothing helps, I thought maybe you could try to do the following:
Build a network dataset in a feature dataset in a file geodatabase just for a minor part of the Washington state. Make sure it did compile and things look good. Then edit the source streets feature class by loading the streets left (you can use Simple Object Loader in ArcCatalog or Load Objects in ArcMap. Then rebuild the network dataset. Since it is just about "re-attaching" the edges, the re-building will go quicker and may not exhaust your resources as it did previously.
Let me know how it goes!