Looks like you need something like Osmosis or imposm. These are tools that will convert raw OSM data output by JOSM into a custom postgis format.
Because OSM is free-tagging, there is no "official" way to convert the XML into a usable data structure that a GIS can read. The main importers currently focus on roads, buildings, points of interest etc. because that is what most mapping currently is. So you kind of have to roll your own if you are looking at something relatively new or exotic, such as indoor building tags. Unfortunately, this is not yet a trivial task.
Finally, there is a suite of tools that will convert OSM data into Spatialite format (which could be a bit easier to get up and running than PostGIS) - but again they will need tweaking for anything custom. There is a network importer, see also http://wiki.openstreetmap.org/wiki/Osm2pgrouting.
Something else I forgot, if you want to query an area you can use Overpass Turbo API. You will probably have to read up on the query syntax, but it will be much easier to get the tags you have put into the database out and export as geoJSON (from where you can use in QGIS for example). You will obviously need to have uploaded your data from JOSM to the OSM database though.