Let $A$ be an $n$-by-$n$ covariance matrix, which must be Hermitian or real symmetric if $A$ is real. It's simple to see that all eigenvalues of $A$ must be real. Suppose $\lambda$ is an eigenvalue of $A$. Then $Ax=\lambda x$ for some non-zero vector $x$. So clearly $x^*Ax=x^*\lambda x=\lambda \|x\|^2$. But since $A=A^*$, we also have $x^*Ax=x^*A^*x=\bar\lambda x^*x=\bar\lambda\|x\|^2$. Hence $\lambda=\bar\lambda$, i.e. it must be real.
The eigenvectors of $A$ (real symmetric) need not all be real. However, it's always possible find a real eigenvector for each eigenvalue of $A$. To see this, suppose $\lambda$ is an eigenvalue of $A$, i.e. $Ax=\lambda x$ or $(A-\lambda I)x=0$ for some non-zero $x$. We've shown that $\lambda$ must be real. This implies that $(A-\lambda I)$ is real and singular. Therefore, solving this real linear set of equations can definitely yield a real non-zero solution for $x$, which is an eigenvector of $A$ corresponding to $\lambda$. (Note that $jx$ is also an eigenvector of $A$ corresponding to $\lambda$, and it's not real.)
Alternatively, suppose $x=u+jv$ is a complex eigenvector of $A$ corresponding to $\lambda$. Then $Ax=\lambda x$ means $A(u+jv)=\lambda(u+jv)$, implying that $Au=\lambda u$ and $Av=\lambda v$, since $\lambda$ is real. So $u$ and $v$ are real eigenvectors of $A$ corresponding to $\lambda$, if they are non-zero. (They can't both be zero.)
In fact, stronger statements can be made. Using the above argument repeatedly in the Schur Triangularization process, we can further show that $A$ can be diagonalized by a real orthogonal matrix, i.e. $U^*AU=D$, where $U$ is real orthogonal and $D$ is real diagonal. So $U$ consists of $n$ real orthogonal eigenvectors of $A$, which diagonalize $A$ and also form a basis for $R^n$.
