A hyper cube with sidelength 1 has a 'volume' of 1.
Then you sorta just define 'volume' (or the proper term 'hypervolume' based on how many hyper cubes you can fit. If the fit isn't perfect, half the size of your cubes, and try again. Whatever limit this system approaches is the true volume. (Edge cases are fractals, and they are already complicated in 3 dimensions)
Edit: each dimension has the same unit length, by definition of that dimension. At least, in 'flat' space.
Distance is just done by Pythagoras. This is actually meaningful because Distance is unique in being preserved by rotations. And rotations are special because they form a subgroup of all linear transformations.
Hence the choice to use Pythagoras for Distance is not arbitrary, it is very natural. Because rotations are very natural.
Edit: each dimension has the same unit length, by definition of that dimension. At least, in 'flat' space. Distance is just done by Pythagoras. This is actually meaningful because Distance is unique in being preserved by rotations. And rotations are special because they form a subgroup of all linear transformations.
Hence the choice to use Pythagoras for Distance is not arbitrary, it is very natural. Because rotations are very natural.