Does a random sample $X_1, \dots, X_n$ imply $n > 1$? I am calculating the variance for an estimator which is a function of a random sample $X_1, \dots, X_n$. However, if $n=1$, then $E(\theta^2) = \infty$ and the variance is infinite. Can I assume $n > 1$ due to the random sample definition $X_1, \dots, X_n$ or does this also include the random sample of {$X_1$}?

It is totally reasonable to get an infinite variance of an estimator, such as the variance of $\bar X$ for a distribution like $t_2$.
Further, estimator variances definitely make sense for samples of just $1$, such as $\text{var}\left(\bar X\right)=\sigma^2$ for $X\sim N\left(\mu,\sigma^2\right)$.
Therefore, this does not seem like a problem. You just have infinite variance when you observe one point.
