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Distance is one of basic physical quantities hence it is "a property that can be quantified". It's basic SI unit is meter. Perhaps it's not much of an answer, but it's as far you can go.
Geometricaly speaking, distance between two places is a function of their coordinates or simply - just a number.
Formula for the function that is used to calculate distance is called metric and can be as simple as Pythagoras theorem in plane or very complicated incorporating also time... (smart enough formula can describe how things fall down, it just needs to give smaller number (distance to the ground) for bigger time).
Distance in physics, is the length of the path (the line or curve) described by an object moving through space. It's unit in Système International d'Unités is metre.
Note: Distance is not the same as displacement. Distance is a scalar and displacement is a vector. To better understand the difference between them you can use the applet from this site: http://www.physics.gatech.edu/academics/classes/2211/main/demos/displacement/DDis.html
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