It depends a bit by what you mean by "modern technology," I suppose, but triangulation is a pretty straightforward tool for determining heights and altitudes. It comes up in navigation all the time, which is why I'm familiar with it. Simple inclinometers like the astrolabe date to classical times; if you know the distance between yourself and and object and can measure the angle between the ground and the top of the object, you then use the Pythagorean theorem to determine the height, expressed as one side of a right triangle. The theorem has been independently invented multiple times.