Expectation Operator on a Matrix

The expected value of matrix $A$, $E(A)$, is just a matrix containing the expected values of the elements of the matrix $A$, as has already been mentioned.
Moreover, a number has no expected value, being a constant. The expression $E(XX^T)$ refers to the $X's$ viewed as random variables, and not as the specific realizations that you have available.

So the expression in the upper most left corner will be $$E(X_0X_0) = E(X_0^2)$$ but you are not supposed to insert in place of $X_0$ the actual value $x_0$ you have available as a realization of the random variable $X_0$. This is a theoretical expression.

If you want to obtain an estimate of these expected values, you have to invoke the covariance-stationarity assumption. When processes are stationary, any continuous transformation or combination of them is also stationary, and also, sample means are consistent estimators of theoretical moments.

Covariance stationarity implies $E(X_0^2) = E(X_1^2)=... =E(X_n^2)$

Define $Z_0= X_0^2$ etc. Then $E(Z_0) = E(Z_1^2)=... E(Z_n^2)$, and the $Z$'s are stationary. If they are stationary, the expression

$$\frac 1{n+1} \sum_{i=0}^{n}z_i \rightarrow_p E(Z_0)= E(X_0^2)$$

and the lowercase letters denote the actual realizations, the data sample you have available.

Moreover, covariance stationarity implies not necessarily that the covariances are constant through time, but that they only depend on the distance between two elements, and not on their exact position in the sequence (i.e. not on the time index). So a priori you assume that, for example, $E(X_0X_1) = E(X_1X_2) =...= E(X_{n-1}X_n)$. If you want to estimate this one-step covariance, by the same argument as before, you can calculate

$$\frac 1{n} \sum_{i=1}^{n}x_ix_{i-1} \rightarrow_p E(X_1X_2)$$

Note that the longer the gap between two variables, the more information from your sample you are going to lose in estimating their covariance.

Finally note that the expression $E(XX^T)$ is the (auto)covariance matrix only if the $X$'s have zero expected value.