As far as I understand, any non-rectifiable curve is not chord arc. So any such curve will do, e.g. the Julia set of $z^2+c$ for $c\neq 0$ in the main cardioid of the Mandelbrot set.
If you want a rectifiable example, if I recall correctly the definition of chord arc requires that the cord between two points has length comparable to their distance. So surely you can just take the graph of a function that oscillates near 0 in such a way that the individual pieces up to a given t will be summable (hence the curve is rectifiable) but the sum is not $O(t)$. For example, $$t+it^{3/2}\sin(1/t)$$ ought to do it if I am not mistaken.