Q: How can I back up my important files when I can't sign in due to disk utility?

If you start in Recovery mode, the Disk Utility available there has the usual File - New -  Disk Image from Folder... command, which should allow you to back up your entire home folder, if that's what you want; or choose sub-folders if appropriate.

OS X: About OS X Recovery

Is Filevault enabled on the HD ?.

Do you have any other bootable backup : a TimeMachine or clone ?

>>I can't sign in due to disk utility?

What exactly happens when you try to start normally ?.

I see. That sounds as if the internal HD can't be accessed at all, & you were automatically started in Recovery mode.

You're unable to mount  (select it; Right click - Mount) then use verify/repair ?

fsck in  Single User mode, occasionally seems to work where DU doesn't :

Resolve startup issues and perform disk maintenance with Disk Utility and fsck

Are you able to select the main HD, or disk0s2 under it, & click the 'New Image' button in Disk Utility toolbar ?. (it's also in the file menu - New - Disk Image from (diskname).

If so, you might be able to backup the whole disk to the external.