Installation of Microsoft SQL Server Data Tools Hangs
I experienced exactly the same error, right down to the pixel.  I figured out the solution by continuing the broken installation using the standard VS 2017 installer (it will recognize that the SSDT installer error'd out).  Resuming this installer reported more information via error logs.  The errors were folder/file access permissions being restricted by McAfee Anti-virus

tldr; disable your anti-virus.