How does this specific set look like? How does the set $  \begin{pmatrix}
        \cos(x) \\
      4\sin(x) \\
        \end{pmatrix} \in \mathbb R ^2$ with $x \in \mathbb R$ look like? 

I guess it should be similar to $f(x) = \arccos(4\sin(x))$ , but none of the programs I know like to plot that
Consider that $\binom{\cos x}{\sin x}$ is the unit circle.  Multiplying the second element by $4$ stretches that circle in the vertical direction by a factor of $4$, so it becomes an ellipse with minor axis $2$ and major axis $8$.

You can note that $\sin(\arccos X) = \sqrt{1-X^2}$, then $4\sin(\arccos X) = 4\sqrt{1-X^2}$.  However, since $\arccos$ has range $[0, \pi]$, this only gives you the upper part of the ellipse.
The topological space is basically homeomorphic to $S^1$