Can I back up and restore my FTL save file?
Solution 1:
On Windows 7, you can find it under My Documents\My Games\FasterThanLight. But only after you've clicked Save+Exit.
But please be aware, that this uncertainty is what makes rogue-like games so enjoyable. I just needed this to copy my career from my laptop to my PC, as the game doesn't seem to support the Steam Cloud.
Solution 2:
In Linux, the savegames are stored in ~/.local/share/FasterThanLight.