Assumption 5 is often listed as a Gauss-Markov assumption and refers to normally distributed error terms in the population. Overall, assumption 5 is not a Gauss-Markov assumption in that sense that the OLS estimator will still be the best linear unbiased estimator (BLUE) even if the error terms are not normally distributed in the population. The fact that OLS estimator is still BLUE even if assumption 5 is violated derives from the central limit theorem, which ensures that when n goes to infinity (basically for very large samples) the estimated coefficients are asymptomatically normal distributed even if the error terms are not.

Overall, OLS estimator remains BLUE even if assumption 5 violated. However, normal distributed error terms in the population come in handy for small sample sizes. That follows from the property of normal distributed error terms, allows hypothesis testing even when the number of observations in a sample is rather small.

Assumptions of Classical Linear Regressionmodels (CLRM)

Overview of all CLRM Assumptions
Assumption 1
Assumption 2
Assumption 3
Assumption 4
Assumption 5