Eigenvalues of orthogonal matrices

The only possible modulus is 1.

Indeed, an eigenvalue is bounded above by any sub-multiplicative norm. But the subordinate to the euclidean norm of such a matrix is 1 so we have that |a|<=1 for such an eigenvalue. Yet, 1/a is an eigenvalue of the inverse of the orthogonal matrix, which is also orthogonal, hence |1/a|<=1 and |a|>=1.

Conversely, the following matrix is orthogonal and has e^ix as an eigenvalue :
$$\begin{pmatrix} \cos x & -\sin x \\ \sin x & \cos x \end{pmatrix}$$