Error in Virtual Box: Failed to open a session for the virtual machine

Download Install OS X Yosemite from the Mac App Store. Use iesd to create a bootable dmg:

iesd -i "/Applications/Install OS X Yosemite.app"/ -o yosemite.dmg -t BaseSystem

Install VirtualBox for Mac Hosts

Create a new VirtualBox guest. Choose Type: [Mac OS X] and Version: [Mac OS X (64-bit)]. Don't choose a specific OS version.

After you create the guest, choose it and click Settings->System->Motherboard->Chipset and choose PIIX3. If you leave the default ICH9 set, the system will never boot. You'll just see a black screen in your guest window.

Choose the yosemite.dmg you created earlier for your guest's optical disk.

Start your guest.

You'll see a wall of text and eventually the OS X Installer screen.

Use disk utility to partition your disk into a single partition. Disk Utility will have proper defaults.

Exit disk utility and run the installer.

You'll find some steps like these below in other places, but they didn't work for me. iesd worked.

 #!/bin/bash  

 # Mount the installer image  
 hdiutil attach /Applications/Install\ OS\ X\ Yosemite.app/Contents/SharedSupport/InstallESD.dmg -noverify -nobrowse -mountpoint /Volumes/install_app  

 # Convert the boot image to a sparse bundle  
 hdiutil convert /Volumes/install_app/BaseSystem.dmg -format UDSP -o /tmp/Yosemite  

 # Increase the sparse bundle capacity to accommodate the packages  
 hdiutil resize -size 8g /tmp/Yosemite.sparseimage  

 # Mount the sparse bundle for package addition  
 hdiutil attach /tmp/Yosemite.sparseimage -noverify -nobrowse -mountpoint /Volumes/install_build  

 # Remove Package link and replace with actual files  
 rm /Volumes/install_build/System/Installation/Packages  
 cp -rp /Volumes/install_app/Packages /Volumes/install_build/System/Installation/  

 # Copy Base System  
 cp -rp /Volumes/install_app/BaseSystem.dmg /Volumes/install_build/  
 cp -rp /Volumes/install_app/BaseSystem.chunklist /Volumes/install_build/  

 # Unmount the installer image  
 hdiutil detach /Volumes/install_app  

 # Unmount the sparse bundle  
 hdiutil detach /Volumes/install_build  

 # Resize the partition in the sparse bundle to remove any free space  
 hdiutil resize -size `hdiutil resize -limits /tmp/Yosemite.sparseimage | tail -n 1 | awk '{ print $1 }'`b /tmp/Yosemite.sparseimage  

 # Convert the sparse bundle to ISO/CD master  
 hdiutil convert /tmp/Yosemite.sparseimage -format UDTO -o /tmp/Yosemite  

 # Remove the sparse bundle  
 rm /tmp/Yosemite.sparseimage  

 # Rename the ISO and move it to the desktop  
 mv /tmp/Yosemite.cdr ~/Desktop/Yosemite.iso 

I had this problem before as well & I need to admit, I just formatted the drive - which is probably the easiest way. If you don't really need any of the old backups, maybe consider this again.

Alternatively you could try to run the Disk Utility App and try to repair your Backup Drive!