I have a network dataset stored in a feature class on one computer. I want to transfer this to another computer but I've tried exporting the geodatabase and copying the geodatabase within the catalog it is contained within and each time I lose the network dataset. The resulting geodatabase just has features stored and no longer within a feature dataset. Is there a way to copy a network dataset over or do I have to copy the files to create it and recreate it in the new location?
You do not say exactly how you are copying the GDB and\or exporting the geodatabase.
You should be able to copy the geodatabase and all its contents should come along.
If this is a file geodatabase, you could try just copying the .gdb with Windows Explorer.
Also you say:
I have a network dataset stored in a feature class on one computerand
The resulting geodatabase just has features stored and no longer within a feature dataset
A network dataset is stored in a feature dataset not a feature class so the first statement seems a bit off, but that might just be simple misstatement. But the fact you say the features are no longer in the feature dataset seems odd. If you copy the GDB in its entirety the schema of the GDB should not change.