Fractals & the intersection of 2 impossible problems

Mathematics is full of interesting problems, some of which are considered to have no solution and are thus 'impossible'. At times, there are 2 (or more) impossible problems that have an overlap due to the specific nature of their construction. One of my favorite examples of this is the overlap between "Squaring the circle" and the "Coastline paradox".

Squaring the circle: As π (pi) is a transcendental number, it is not truthfully possible to draw a square with the same area as a circle; although close approximations are possible. Likewise, it is not possible to "circle the square".

Coastline paradox: The length of any segment of coastline is technically impossible to measure, you can only use approximations. The problem here, is that the complexity of the measurement increases with a decrease in measurement scale.

As a thought experiment, "squaring the circle" is indeed impossible, if you are dealing with perfectly straight lines, such that they are perfectly straight one-dimensional features in some normal Euclidean space. However, could you construct a 'square' with lines of fractal-complexity, such that the bounded area converges along the same path with respect to π? More importantly, would such a fractal need to be self-similar at each scale (replication) or would some other evolutionary symmetry be needed?

While my own interests often trend towards the 'applied', it is fun to occasionally delve into abstract and theoretical concepts. That said, fractal analysis does cross the street of theory & applied quite frequently.