Conceptually you can't estimate the effect of a variable if it isn't... a variable. More concretely the intercept term and the "slope" for X1 cannot be uniquely identified from the data because they're perfectly collinear.
An interesting starting point for you for you to mull over. Effectively if you run OLS and want an estimate of the intercept, that's exactly what you have. The data matrix has x_0 with all the values = 1.
From perspective of G-M assumptions, this is likely to violate the collinearity assumption. Here is an amazing read for anyone looking into OLS assumptions: Berry, W. D. (1993). Understanding regression assumptions (Vol. 92). Sage Publications.