I couldn't find OALD, but I believe that the online Oxford Dictionary should be close enough. A similar definition for level out/off can be found under verb sense 2.
2 [no object] (level off/out) begin to fly horizontally after climbing or diving:
he quickly levelled off at 1500 ft
- (of a path, road, or incline) cease to slope:
the track levelled out and there below us was the bay
- remain at a steady level after falling or rising:
inflation has levelled out at an acceptable rate
This definition is for level out being used as an intransitive verb (denoted by [no object]), which is quite different from your quoted sentence:
"Seems like society leveled itself out once again."
where sense 1 (give a flat and even surface to) and sense 3 (make [something, especially a score in sport] equal or similar) are more relevant.
Thus, the phrase society leveled itself out should not be understood as the society has changed its direction to continue at a steady level. (It might be possible if it were about the future direction of economy.) In this context, it just means: everyone has an equal chance.