high: standing up well above whatever surrounds it, as a mountain, a wall, or the height where God is enthroned
The picture behind this word is simple elevation: something that rises well above the level of whatever is around it. In the Gospels it is almost always a mountain, the towering kind a person has to be led up, and in the vision of the new Jerusalem it is a city wall so tall that its height is worth reporting. Nothing in the word itself says the thing is sacred or impressive. It only says that the thing is up there.
Greek could also use the adjective on its own, with no noun attached, as the name of a place. When Hebrews says the Son sat down "in the heights", the word for high is doing the work English does with the word "heaven": the high place is where God's majesty is. The same move stands behind the comparison in Hebrews 7:26, where the priest we have has been made higher than the heavens themselves.
One use comes straight out of the Greek Old Testament. There God brings Israel out of Egypt "with a high arm", an arm raised and ready, the way anyone raises an arm to strike or to lift. Paul's hearers in the synagogue at Antioch would have caught the phrase at once as the language of the exodus. It does not mean that God's arm is tall. It means his power was out in the open where everyone could see it.
Greek also has a noun for the height itself (ὕψος) and a verb for lifting something up (ὑψόω), and it saves the superlative (ὕψιστος) for the very top: the highest heavens, and God as the Most High.