I have done the conversion CAD -> GIS several times, unfortunately, to your request, I performed with the usage of ArcGIS. However, I would like to highlight, that there is always a part left for a manual work due to the difference between GIS and CAD and humans factor in it. At least in my case, I worked with data that was acquired through the surveying.
The last time I did it, I have used a feature from ArcGIS toolbox, known as
ArcToolbox > Conversion Tools > To Geodatabase (CAD to Geodatabase)
I had lots of buildings, therefore some of them were not converted into proper polygons what they should be. Because of that, I used a function to transform unclosed polygons, i.e. polylines into polygons.
ArcToolbox > Data Management Tools > Features > Feature to Polygon
But you have to be careful since some of the polygons cannot be produced because of the distance between starting and ending point of the polyline.
Additionally, I can refer you to Converting polyline to polygon in ArcGIS Desktop for floodplain map? and How to close polylines that mark shape of cirque glaciers?.
With the development of QGIS, I can recommend the most common way, which was mentioned by Erik, to use
Layer > Add Layer > Add Vector Layer > AutoCAD DXF (*.dxf, *.DXF)
Alternatively, there is a bunch of plugins in QGIS, e.g. CADDigitize, Another DXF Importer / DXF2Shape Converter, CadInput etc. Can refer you to DXF Import QGIS: Text Components.
Not sure about ogr2ogr. Perhaps HeikkiVesanto is correct.