This is the surprisingly the first time I've come across this. If I wanted to export a line shapefile from ArcGIS to another database (say Access) would I first need to convert the vertices to points and add the x y columns? To then reuse the data in a GIS I would need to convert back to lines. I feel sure I must be missing something.
If you use a spatially enabled database, you can simply and safely store the lines and edit them as well within the RDBMS. Examples of databases that are spatially enabled are MS SQL Server 2008 and above, PostgreSQL with PostGIS, or MySQL with spatial data extensions.
Access is not a spatially enabled database, but you can still store shapefiles (points, lines and polygons) within Access as a geodatabase.
Warning: If you decide to store your data in MS Access, do not use Access to delete features, and avoid editing the attribute tables through Access. Doing so risks corrupting your data and making it unusable.
No, you shouldn't have to convert to raw coordinates if you are importing shapefiles into a database that understands spatial data. For example, Postgis provides shp2pgsql to load shapefiles into Postgres, storing the shapes as GEOMETRY datatypes.
If the target database does not have support for geometries, you will have to convert the shapes to some format it does understand. Though it's been many years since I've used Access, I assume it does not have any geospatial datatypes. In this case, a better strategy would be to take advantage of various textual representations that are easily generated, stored, and converted. KML, GeoJSON, WKT are all easily generated using either ArcMap or OGR and could be stored as text in any database.