The geometric network is NOT part of the ArcGIS Network Analyst extension. It is part of the core ArcGIS, so you don't need any extensions to build or analyze a geometric network.
To build a geometric network, you need at least an ArcEditor license level. To analyze the geometric network, all you need is the ArcView license level. See the ArcGIS Desktop Functionality Matrix for full details -- geometric networks are listed in the fourth column from the right. This matrix is specific for ArcGIS 10, but the licensing of geometric networks is the same in ArcGIS 9.x as well.
In ArcMap, there is a toolbar you use for analyzing geometric networks; it is named "Utility Network Analyst" and is described in the ArcGIS Desktop Help here. This toolbar is part of core ArcGIS Desktop and does not require any extensions.
Be sure that your line features edited in such a way that they are snapped to the node -- the Connect Feature tool on the "edit GN toolbar" only works when the endpoints (or vertices) of the features you want to connect are already coincident. Once the vertices are coincident, then the features should automatically connect to each other (you should only need to use the Connect Feature tool if they are indeed coincident but still not connected). Often times the lines appear to be coincident when you're zoomed out, but when you zoom in you will see that they are not. Also, be sure that you are using the correct edge feature type (simple edge vs. complex edge) -- simple edge features can only connect to other features via its endpoints, while complex edge features can connect mid-span. If you would like someone to examine your data in more detail, you can contact Esri Technical Support and send it to them.