In ArcMap it is possible to export a dxf into shapefile. In the TOC right-clic on dxf layer and in the menu select Data>Export Data. In the following windows, choose shapefile format.
You might use the 'Conversion Toolbox"; "Import from CAD (conversion)". Have a look at the ArcGIS web help http://resources.arcgis.com/en/help/ , or, e.g., http://resources.arcgis.com/en/help/main/10.1/index.html#//001200000024000000
You can use ArcToolbox to import from CAD, and more, automatize all operations of construction of topology to make it easy to exploitation in GIS format
I don't see an 'Import CAD' option in ArcGIS 10.5.1; only 'Export to CAD'. I tried using the Data Interoperability Quick Import tool, but got an error. Not even FME would read my DXF file, but imported into QGIS no problem, and from there I was able to save in a different format (SHP).