Consider a standard oneway ANOVA model given by $$ Y_{ij}=\theta_i+\varepsilon_{ij} $$ for $i=1,\ldots,k$ and $j=1,\ldots,n_i$, where $\varepsilon_{ij}$ are iid $N(0,\sigma^2)$ random variables. The residuals are defined as $$ e_{ij}=Y_{ij}-\frac1{n_i}\sum_{k=1}^{n_i}Y_{ik} $$ for $i=1,\ldots,k$ and $j=1,\ldots,n_i$. The variance of the residuals is given by $$ \operatorname{Var}e_{ij} =\sigma^2\Bigl(1-\frac1{n_i}\Bigr). $$ This means that the variance of the residuals increases if we increase the sample size $n_i$. Since the residuals $e_{ij}$ essentially estimate the errors $\varepsilon_{ij}$, I would expect the variance of the residuals to be larger than $\sigma^2$ and get closer and closer to $\sigma^2$ as the sample size $n_i$ increases. So the expression of the $\operatorname{Var}e_{ij}$ seems counterintuitive to me.
Why the variance of the residuals increases as the sample size $n_i$ grows?
Any help is much appreciated!