Following the usual definition of length of a curve, the curve have to be rectifiable to have finite length. However for example the curve
$ \gamma \colon [0,1] \to \mathbb{R}, t \to t \cos^2(\pi/t) $
is not rectifiable. However I would say it makes perfectly sense to say it has length $1$.
So my question is, if there is a more general definition of length of a curve such that my example above has length $1$ (however such that for example the graph of the function $\gamma$ above has infinite length).