Unable to boot - facing stop sign; Virtual Whole Storage needs repair; Time-Machine cannot find restoration disk destination
You probably repartitioned the main volume the wrong way. At least your CoreStorage Logical Volume Group got corrupted either by partitioning or by "repairing".

The ultimate method to reset the drive is to completely destroy the GUID partition table of the main internal disk. This will render all data on the disk useless and you need an up-to-date backup to restore your old volume.

To destroy the GUID of your internal physical disk you have to boot to an external boot volume. This may be a OS X installer thumb drive or Internet Recovery Mode (which net-boots to a recovery image provided by Apple over the network). You mustn't boot to Recovery Mode which loads a basic system in a mounted but compressed recovery image in your recovery partition diskXs3).


Detach any external disks - especially your Time Machine backup drive
Boot to Internet Recovery Mode
Open Terminal in the menubar Utilities -> Terminal
Get an overview (especially the gpt command is important!):

diskutil list
gpt -r show disk0
gpt -r show disk1


Check which of both disks is your internal disk. Usually the internal disk is disk0 and the recovery volume is disk1 if you have only one disk in your Mac. In rare cases disk1 is your internal disk and disk0 is the recovery volume. The recovery disk has ~ 2600000 blocks (or ~1.3 GB). Even the smallest SSD is much larger. Below I assume your internal disk has the disk identifier disk0 - so use disk1 (or disk2, disk3) instead of disk0 in the commands if you find an other disk identifier for your internal physical disk.
Unmount disk0:

diskutil umountDisk /dev/disk0

Delete the GUID:

gpt destroy disk0


If this doesn't help, then the hard way:

dd if=/dev/zero of=/dev/disk0 bs=512 count=40

Create a new  GUID partition (this also creates a fresh pMBR):

gpt create -f disk0

Enter exit and quit Terminal.
Open Disk Utility and partition your internal disk: one or two partitions and at least one JHFS+ volume.
Quit Disk Utility
Attach your Time Machine backup drive, open Restore Time Machine backup (or similar) from OS X Utilties and restore your backup.




This usually won't restore your Recovery HD. You have to do this either manually with Recovery Partition 4.0.4 or by reinstalling your current system with an Apple system installer (i.e. "Install OS X Yosemite" 10.10.5, which usually won't affect your data)