I'm not entirely sure either, but maybe it's meant to be interpreted in light of the 'existence precedes essence' concept of existentialism. 'Being' might then mean leading a life where one choses their destiny, in contrast to mere 'existence'. Maybe the translation from French is not perfect?

"The paradox of human life is precisely that one tries to be and, in the long run, merely exists." I think what she means is: if you have your life ahead of you, you can still choose your destiny (you can be something). However, having lived that life and looking back at it, history is fixed and there is nothing to choose when it comes to the past (it merely exists).