I am doing an assignment where I calculate the consumption function of a certain country from empirical data and I am getting a negative intercept. How can this be interpreted because usually, isn't the intercept always positive?
I assume that you ran a linear regression to find the $C(Y)$ function and by negative intercept you mean $C(0)<0$. This does not contradict anything because what you have is an estimate of the $C(Y)$ function based on empirical data that was probably centered around some non-zero value of $Y$. So it is quite possible that the function's estimate is incorrect far from the observed range of $Y$, especially if you were using a linear regression which is a very basic functional form.
If the actual $C(Y)$ consumption function, not the estimate, were such that $C(0) < 0$ then indeed we would have a contradiction. But as long as you are clear that what you have is an estimate for $C(Y)$ this problem does not arise.