How do I calculate $\sigma^2$ within the $\varepsilon = N(0, \sigma^2)$ notation The error term for an AR(1) model is assumed to be $\varepsilon = N(0, \sigma^2)$, a mean of 0. An example of a source that mentions this:

The AR(1) model can be written in intercept form,
$$z_t = \beta_0 + \beta_1 z_{t-1} + a_t$$
where $t = 1, \dots, T$, $\beta_0$ is the intercept coefficient, $\beta_1$ is the slope coefficient and $a_t$ is assumed to be $\text{NID}(0,\sigma_a^2)$.

Additionally, the error term is also defined as having a constant variance, in other words, the sigma squared isn't a variable but a constant.
My question is how can I calculate the sigma squared/variance of the error term.
I assume the variance isn't the same for each data set, that it's rather dependent on the data set and that I need to do some calculation with the data that I have to find the specific variance.

Typically, the values $\beta_0$, $\beta_1$ and $\sigma$ would be the unknown parameters in your model.  So you don't calculate them --- you use the data to make inferences about them (e.g., via estimators).
