Hint:
$$\operatorname{var}\left(\sum_{i=n}^n X_i\right)
= \sum_{i=1}^n \operatorname{var}(X_i)
+ \sum_{i=1}^n \sum_{j=1; j \neq i}^n \operatorname{cov}(X_i,X_j)\tag{1}$$
How many terms in that double sum on the right are nonzero?
What is the maximum value that these terms can have?
The question is very poorly formulated and both parts do not necessarily have
an answer.
Assume that $n$ is a fixed number.
We are given that $X_i, 1 \leq i \leq n$ are dependent random variables
such that each $X_i$ is correlated with $2$ other $X_j$ but uncorrelated
with all the others. Furthermore, both of the two nonzero covariances
$\operatorname{cov}(X_i,X_{j_{1}})$ and $\operatorname{cov}(X_i,X_{j_{2}})$
are no larger than $n^{-2}$. (Curiously, this upper bound on the covariances
does not involve the variances of the random variables at all, and thus seems
more like a bound on the (Pearson) correlation coefficients).
So the $n\times n$ covariance matrix
$C$ with entries $C_{i,j} = \operatorname{cov}(X_i,X_{j})$ has
three nonzero entries in each row: one entry (on the diagonal)
being a variance and the other two off-diagonal entries being at most
$n^{-2}$.
Can such a covariance matrix exist? Yes, for example, suppose that
only the following pairs of random variables are correlated:
$$(X_1,X_2),~ (X_2,X_3),~ (X_3,X_4),~ \ldots,~ (X_{n-1},X_n),~ (X_n,X_1).$$
The covariance matrix in this case is a tri-diagonal matrix with two
additional nonzero entries at $(n,1)$ and $(1,n)$. Note, however,
that this correlation structure cannot be expanded to include
a $(n+1)$-th random variabe $X_{n+1}$.
So, applying what was hinted at above in $(1)$, we get that
$$\begin{align}
\operatorname{var}\left(\sum_{i=n}^n X_i\right)
&= \sum_{i=1}^n \operatorname{var}(X_i)
+ \sum_{i=1}^n \sum_{j=1; j \neq i}^n \operatorname{cov}(X_i,X_j)\\
&\leq \sum_{i=1}^n \operatorname{var}(X_i) + 2n\cdot n^{-2}
&{\scriptstyle{\text{only}~2n~\text{terms are nonzero}}}\\
\operatorname{var}\left(\sum_{i=n}^n X_i\right)
&\leq \sum_{i=1}^n \operatorname{var}(X_i) + \frac{2}{n}\tag{2}\\
\operatorname{var}\left(\frac 1n\sum_{i=n}^n X_i\right)
&\leq \frac{1}{n^2}\sum_{i=1}^n \operatorname{var}(X_i) + \frac{2}{n^3}\tag{3}
\end{align}$$
and if all the random variables have the same variance $\sigma^2$, then
$$\operatorname{var}\left(\frac 1n\sum_{i=n}^n X_i\right)
\leq \frac{\sigma^2}{n} + \frac{2}{n^3}.\tag{4}$$
This result is also found in @AlecosPapadopoulos's answer to this question.
Returning to $(1)$, note that variances are nonnegative numbers, and thus
it must be that
$$\sum_{i=1}^n \sum_{j=1; j \neq i}^n \operatorname{cov}(X_i,X_j)
\geq -\sum_{i=1}^n \operatorname{var}(X_i).\tag{5}$$
Since there are only $2n$ nonzero terms on the left side of $(5)$,
the average of the $2n$ nonzero covariances is bounded below
by the negative of half the average variance. For the special
case of all variances being $\sigma^2$ and all the nonzero
covariances being equal, we have the peculiar result that
$$-\frac{\sigma^2}{2} \leq \operatorname{cov}(X_i,X_j) \leq \frac{1}{n^2}.$$
Things are equally strange when one tries to derive a Law of Large
Numbers type result in this schema and so we are looking at
increasing values of $n$ instead of just a fixed integer value.
Assume without loss of
generality that $X_1$ and $X_2$ are correlated, and thus
$$\operatorname{cov}(X_1,X_2) \leq \frac{1}{n^2}.$$
Since this must hold whether we are looking at $n=10$ or $n=100$
or $n=1000$ etc., we conclude that the upper bound on the
nonzero correlations must be $0$, and since these covariances
are supposed to be nonzero, it must be that all the nonzero
covariances are strictly negative numbers. The alternative,
that the $X_1$ and $X_2$ in the list $X_1, X_2, \ldots, X_{10}$
are not the same as the $X_1$ and $X_2$ in the list
$X_1, X_2, \ldots, X_{100}$ is too dreadful to contemplate.
There is also the issue of what the covariance matrix looks like
since the structure described above cannot be extended. As
noted in Alecos's answer, we need to modify the statement that
"each random variable is correlated with exactly two other
random variables" to something like "each random variable
(with the possible exception of one or two) is correlated
with exactly two other random variables". And of course,
in the absence of any information about the variances
of the random variables, we cannot guarantee any kind of
law of large numbers result. $\operatorname{var}(X_i)$
being finite for all $i$ is insufficient; we need the
variance of the sample mean to approach $0$ as $n \to \infty$.