What does $\textstyle y \in \Re^{100}$ mean?

I believe this means that $y$ is a $100^{th}$ dimensional real vector. From Wikipedia

The notation $R^n$ refers to the cartesian product of $n$ copies of $R$, which is an $n$-dimensional vector space over the field of the real numbers; this vector space may be identified to the $n$-dimensional space of Euclidean geometry as soon as a coordinate system has been chosen in the latter. For example, a value from $R^3$ consists of three real numbers and specifies the coordinates of a point in $3$‑dimensional space.

This may or may not be what Brevan Ellefsen has in mind, but if we say, for instance, that $y \in \mathbb{R}^2$, we often mean that $y$ is an ordered pair of real values: that is, $\{(p, q) \mid p, q \in \mathbb{R}\}$. Thus, $(1, 4), (\pi, \sqrt{2}) \in \mathbb{R}^2$, but $(i, 5), (\text{Bert}, \text{Ernie}) \not\in \mathbb{R}^2$.

In that light, we might guess (depending on context) that $\mathbb{R}^{100}$ is the set of all ordered tuples of real values containing $100$ elements. But without more context, it's hard to be sure.

In set theory, we write $X^Y$ to denote the set of all functions from $Y$ to $X$. This makes sense with the notations you know, for instance $\mathbb R^2$ is the set of functions from $\{1,2\} \to \mathbb R$ ; we usually denote such a function by its values, that is, $(x_1,x_2)$ (the function would be $x : \{1,2\} \to \mathbb R$ and $x(1) = x_1$, $x(2) = x_2$). I think the only unconventional thing about your question is the choice of calligraphy for the set of real numbers ; unless it's really some set described in your book, the real numbers are usually typed with mathbb so that it comes out like this $\mathbb R$. So $y \in \mathbb R^{100}$ just means a vector of real numbers with a hundred coordinates.

Hope that helps,