Backup files from Terminal in Recovery Mode
My goal was to copy files to the external hard drive. They are stored in Macintosh HD (I originally thought I will find Macintosh HD in a subfolder of OS X Base System - wrong assumption). There was no Macintosh HD in Volumes folder, because I first needed to unlock it (I use FileVault).

First you need to find the lvUUID of your Macintosh HD. Use this command:

diskutil list // lvUUID


Look for text looking similar to this: 

Logical Volume on disk0s2 
4B2EFAAE-C871-4E6D-AB15-2DDE604B97CE // this is lvUUID
Unlocked Encrypted


To unlock use this command:

diskutil cs unlockVolume lvUUID #replace lvUUID by the one shown in the diskutil listing


You will be prompted with password for your user account.
After that Macintosh HD will be present in /Volumes/ folder.

All thanks to klanomath for enlightening me. See full Q&amp;A here: No Macintosh HD in Volumes folder