Backing up Macintosh HD via Disk Utility (error -5344)
If you have two Machintosh HDs together, choose the second (indented) one. You may need to unlock it via the context menu.

Then tell Disk Utility to create an image of the volume directly. In Disk Utility with the Macintosh HD volume selected, use New disk image on the toolbar or select New > Disk Image from "Macintosh HD" from the File menu.

Bear in mind that the created image has all of the problems from the original filesystem, and the creation of the image may fail if there are underlying I/O errors.

You could instead install the OS on the external drive and manually copy what you need to keep. If you get an error (usually -36 or -50), divide the problematic set in half and keep copying until you've either found a set of files you don't care about or narrowed it down to the exact things that have errors and have to be skipped.

If you need applications that weren't just dragged to /Applications, use the migration assistant, and hope it doesn't hit something that won't copy.