When a definition applies to a much larger class of spaces $\ X\subseteq \mathbb R^3,\ $ then such definition rather is not directly related to any parametrization for the smaller class of the parametrized curves (which can be rectified). Several of variants of Hausdorff dimension for dimension 1 would be a possible answer. Another could be
$$\lim_{r\rightarrow +0}\ \frac {V(X+B(r))}{\pi\cdot r^2}$$
where $\ X+B(r)\ =\ \{y\in\mathbb R^3:\ \exists_{x\in X}\ d(x\ y)\le r\},\ \ d\ $ is the euclidean distance, and $\ V\ $ stands for volume. This will certainly work properly for the piecewise $\ C^1$-curves, but most likely for all rectifiable curve; and of course--by definition--for many other spaces in the Euclidean space. Again, variations of this definition are possible, e.g. using cubes with edges $\ \left[\frac k{2^n};\frac{k+1}{2^n}\right],\ $ etc.