Thinking on a specific problem, led me to a more general question. Thus, here are two questions:
1) As the title suggests, does there exist a r.v. $X$ with $E[x]= \mu$ and $Var[X]=\sigma^2$ for some fixed $\mu$ and $\sigma$. What are the restrictions that need to be imposed on $\mu$ and $\sigma^2$?
2) Now consider the following problem: Is it possible to give an example of a sequence of random variables $(X_n)_{t \in T}$, where, say, $T= \mathbb{N_0}$ or $T= \mathbb{Z}$, such that $$Cov(X_t,X_{t+h})= \begin{cases} 1, & \text{if $h=0$} \\ 0.4, & \text{if $|h|=1$} \\ 0, & \text{otherwise.} \end{cases}$$
As I said in the comments, for the first question you can just take $X\sim\mathcal N(\mu,\sigma^2)$ if $\sigma^2>0$, $X\equiv\mu$ if $\sigma^2=0$, and otherwise it is impossible. For the second question, recall that given $R:T\times T\to\mathbb R$ there exists a mean-zero Gaussian process $(X_t)_{t\in T}$ such that $E(X_sX_t)=R(s,t)$ if and only if $R$ is symmetric and positive definite, that is, $R(s,t)=R(t,s)$ and $\sum_{i,j=1}^nR(t_i,t_j)x_ix_j\ge0$ for all $t_1,\ldots,t_n\in T$ and all $x\in\mathbb R^n$. In your case this follows from e.g. the fact that $R(t,t)>0$ and $\sum_{s\neq t}R(s,t)<R(t,t)$ for all $t$.