When calculating the dimension of a fractal shape, using the intersecting boxes method (where the number of intersecting boxes at different scales is compared; this is described at the starting at around the 10:30 mark in this video), boxes within an enclosed area of the fractal are not counted.
This makes me think of fractals as structures made of lines of 0 areas. Simplistically, I think of them as hollow structures made out of lines. Does this make mathematical sense? If so, how can some fractals have a finite or infinite area or volume? I can logically comprehend an infinite length/perimeter, however, area and volume make no logical sense to me.
Please note that my math knowledge maxes out at calculus. Would love to hear some logical explanation!
The video describes a technique for computing the box-counting dimension of a bounded subset of a Euclidean space (e.g. $\mathbb{R}^d$ for some nonnegative integer $d$). For a reasonable definition of "hollow", all sets with non-integer box-counting dimension will be hollow. This follows from the observation that if a set $E \subseteq \mathbb{R}^d$ contains an open set $U$, e.g. if there is some point $x \in E$ and some number $r > 0$ such that $$ B(x,r) := \{ y\in\mathbb{R}^d : |x-y| < r \} \subseteq E, $$ then we will have $$ \dim(E) = d, $$ which is an integer. This implies that any set with non-integer box counting dimension must have empty interior, in the sense that it cannot contain any open sets. Having no interior feels like a good notion of "hollowness", which gives an answer to your question.
Alternatively, if a subset of $\mathbb{R}^d$ has non-integer box-counting dimension, then it must have zero $d$-dimensional Lebesgue measure (essentially, the Lebesgue measure generalizes the idea of area or volume). In this sense, a "fractal" has no mass, and is therefore "hollow". Again, this gives an answer to your question.
That being said, your definition of "fractal" is essentially tailor made to get this answer, and leaves out a lot of subtleties. For example, I think that most working mathematicians would regard a fat Cantor set as fractal. However, fat Cantor sets have positive measure, which implies that they have dimension 1 (i.e. the largest possible dimension which they could have). Fat Cantor sets still have empty interior (so maybe you would consider them hollow?), but have positive Lebesgue measure, which means that they do have "mass". Should they be considered hollow?
Finally, consider the following: if $C$ is the usual ternary Cantor set, then $$ E^d := C \times [0,1]^{d-1} \subseteq \mathbb{R}^d$$ (the Cartesian product of $C$ and the unit square in $\mathbb{R}^{d-1}$) has non-integer box-counting dimension, but also contains line segments, squares, cubes, etc. of every dimension less than $d$. Such a set is not made up of lines of zero area.