Every time I open Blender I want to cry. This time I struggled through the interface and found the way to export the raw undeformed models from the big buck bunny characters .blend files.
After finding it, the solution is simple:
File > Export > Wavefront (.OBJ)
Then on the left side under "Export OBJ" uncheck "Apply Modifiers" and uncheck "Selection Only" and "Animation". Then save.