How to extract files from a broken non-bootable OS X without reinstalling OS X? What can I download/install onto a USB drive, have it boot off that USB drive and then copy the files I need to another USB flash drive?

The error message I get when booting is "Missing Operating system". I cannot reinstall OSX currently (no media for it).

The Mac in question is an iMac so I cannot easily take out the hard drive.
Did you try...
Internet Recovery ?
before resorting to
Target Disk Mode or
How to install OS X on an external drive connected to your Mac  or
Create a bootable installer for OS X Mavericks or Yosemite
- which, as already mentioned, needs another Mac