Error term in linear regression

I'm reading about a linear model which is fit to an equation, $Y = \beta_0 + \beta_1X + \varepsilon$, where $B_0$ is the intercept, $B_1$ the slope, and $\varepsilon$ the error term. My question is, when $B_0$ and $\varepsilon$ are both constant, don't they reduce to a single term? Why is there a separate term for the intercept and the error?

Not leaving out subscripts would help. The equation is $y_i=\beta_0+\beta_1x_i+e_i$, where one intercept is estimated for the full sample but the residuals $e_i$ vary across observations.
If you had that you would have a deterministic model. You are right that they would comprise a single term, but $\varepsilon$ isn't constant; it is assumed to have constant variance. This is called the assumption of homoscedasticity. A more appropriate and complete way to write your model is:
$$Y = \beta_0 + \beta_1X + \varepsilon \\ \text{where }\varepsilon\sim\mathcal N(0, \sigma^2_\varepsilon)$$ (Note that the solution to this depends on whether you interpret your model as relating the random variables or as describing a specific observation. I interpreted your case as the former; @PatrickCoulombe interpreted your case as the latter. I think both are—at least potentially—correct.)