Woodie is the last character you unlock through earning XP in Sandbox Mode. Every day you survive in Sandbox earns you 20 XP to Unlock Woodie you need a total of 1600 XP that equates to 80 days. You don't need to survive 80 consecutive days in one play-through every time you die so long as you made it one day you get 20 XP. Something to note is days spent in Adventure Mode does not count towards earning XP.
Source: Don't Starve wikia
http://dont-starve-game.wikia.com/wiki/Experience