For "nothing to do with cardinality", the Cantor set has the same cardinality as the reals, and yet has measure zero. For arbitrarily small size, let's start w

For "nothing to do with cardinality", the Cantor set has the same cardinality as the reals, and yet has measure zero. For arbitrarily small size, let's start with an enumeration of the rationals. Now pick ε > 0. Let's put an open interval of size ε/2 around the first rational, ε/4 around the second, ε/8 around the third and so on. The union of those intervals is an open set of length bounded above by ε/2 + ε/4 + ε/8 + ... = ε. (Note, it is actually smaller than this because some of the intervals overlap...) That constructs an open set, which includes every rational, of size as small as we like. So "cardinality" and "the measure of a set" have no particular relationship, other than that the measure of a countable set is always 0.