For the sake of completeness, here's what representation theory has to say (although since the relevant algebra is commutative it really becomes commutative algebra). You want to study finite-dimensional representations of (finitely-generated modules over) the algebra $R = \mathbb{R}[x]/(x^n + 1)$. Now, $\mathbb{R}[x]$ is a principal ideal domain, so by the structure theorem any such representation decomposes into a finite direct sum $\bigoplus (R/f_i(x))^{e_i}$
where $f_i(x)$ is an irreducible factor of $x^n + 1$ over $\mathbb{R}$. (One can also deduce this using Jordan normal form, but this is a special case of the structure theorem anyway.) Now, the identity $x^n + 1 = \frac{x^{2n} - 1}{x^n - 1}$
shows that the roots of $x^n + 1$ are the $2n^{th}$ roots of unity which are not $n^{th}$ roots of unity; these are precisely the roots of unity of the form $e^{ \frac{\pi i k}{n} }$ with $k$ odd. If $n$ is even, these come in complex conjugate pairs, and so all the $f_i$ are quadratic; if $n$ is odd, $x^n + 1$ has a unique linear factor $x + 1$ and the remaining $f_i$ are all quadratic.
Hence if $n$ is odd the finite-dimensional representations can have any finite dimension, and if $n$ is even the finite-dimensional representations can have any even dimension.