Can a closed rectifiable curve be "bad" in all directions? How can one tell whether a point $P$ not on a closed rectifiable curve $C$ is inside or outside $C$? 
If $C$ is piecewise smooth one can throw a ray $R$ from $P$ in a random direction and count the number of intersections between $C$ and $R$. Odd - $P$ is in; even - $P$ is out. If the a line segment of $C$ lies on $R$ one can either collapse it into a single point.
Now, if $C$ is merely rectifiable then it's not clear that the number of intersections between $C$ and $R$ is going to be finite. Thus the questions:
Is it possible for a rectifiable curve $C$ and a point $P$ be in such position that for a ray $R$ the number of intersections between $C$ and $R$ will be infinite (apart of line segments of $C$ lying on $R$)?
Is it possible for the above travesty to happen for every ray $R$ from a fixed point $P$?
 A: A candidate curve  with few directions with finite number of intersections might constructed from the Blancmange curve as a fractal curve, that can serve as a building block for constructing "evil" Jordan curves with Hausdorff dimension 1 (for the properties of the more general Takagi functions refer to e.g. this article).  
The simplest idea would be to take the union of the curve itself and its reflection at the x-axis (refering to the images on the linked webseite).  
A bit more complicated is the idea of constructing an analoge of a "sine wave" by point-reflecting the curve at one of its endpoints and use a phase shifted version of the result as an analogue of a "cosine wave". A parametric function $x(t)=cosinewave(t),\ y(t)=sinewave(t)$ may be even better, but I haven't done any investigations on whether the set of inner point of the region bounded by that "Blancmonge circle" is connected.  
Caveat:
But, despite having Hausdorff dimension 1, the resulting curves are not rectifiable and thus not an example of the sought kind of curves.
