How to copy files on Mac that fails to boot?
I solved this using Mac Os X Lion install DVD. 


Boot your Mac from Mac Os X Lion install DVD by holding Option key at startup
When a window appears go to menu and find Terminal and run it
From inside terminal you can copy any files ( I used commands: df, ls, cd, mv )
This works for Macbook Pro late 2011 but will not work for 2012 and later models because Apple prevented them from being able to boot from install DVD or USB. Thus all recovery tools like DiskWarrior Boot DVD, Drive Genius and similar will not work on 2012 and later macs. 

The other option is to ask a friend with Mac to make a Recovery USB using OS X Recovery Disk Assistant v1.0 and make a DMG file from it using Disk Utility and send you over internet. Then use the TransMac application under Windows and burn DMG on your USB stick. You will be able to boot from this USB stick into Recovery Mode and from there you can run Terminal to copy\move files or do whatever you need.


Apple should distribute some bootable ISO or DMG image which can be downloaded from their website and burned on disk or USB and used to recover Mac offline.