The root meaning of holiness, k’dushah, is probably “separate”.
The word is used for marriage – kiddushin – implying that the spouses are separated from the rest of the community in order to be special to each other. Shabbat is ushered in by Kiddush, proclaiming that the day is separated from the rest of the week. God is K’dosh Yisra’el, The Holy One of Israel, because He is a Being apart from every other.
So when the Torah calls us a people that is kadosh, it means that we are a people with special traits and teachings.
But in addition to separateness, k’dushah indicates spirituality. We are not just different because of our history and culture, but because we have – or ought to – a dimension of spirituality as d’vekim baShem (Deut. 4:4).
The Hebrew means clinging or cleaving to God. Devek in modern Hebrew is glue; d’vekim baShem means to adhere to God.
It was never easy to adhere to God – there were times when He was tempted to dislodge us and times when we wanted to be free from Him. Neither side ever abandoned the other.
As far as the Jewish people are concerned one of the many Jews who suffered torments as a Jew is said to have told God, “Look, God, You are doing all You possibly can to make us abandon You… but I give You notice, God: it will not help You in the least. You can try us with every kind of suffering, but nothing will induce us to leave You!”