Cantor set: Start with [0,1]; remove the middle third; repeat. After n steps, 2ⁿ intervals remain, each of length (1/3)ⁿ → total measure = (2/3)ⁿ → 0.
Yet the Cantor set is uncountable (has cardinality of the continuum) and has Hausdorff dimension log2/log3 ≈ 0.631.
Devil's staircase (Cantor function): F(x) is constant on each removed interval, continuous, F(0)=0, F(1)=1, yet F'(x)=0 almost everywhere — "all the increase happens on a set of measure zero."