The Gauss-Markov theorem states that for a linear model
$$y = X \beta + \epsilon $$
if both of the conditions are true
$$\operatorname E[\epsilon \mid X] = 0$$ $$\operatorname{Var}(\epsilon) = \sigma^2 I < \infty $$
then the standard OLS estimator $(X'X)^{-1}X'y$ is the best linear unbiased estimator.
Now suppose we measure $X$ with errors. Then we have
$$y = (X + \mu)\beta + \epsilon = X\beta + \mu\beta+\epsilon$$
If $\mu$ is of mean $0$ with constant variance, both assumptions still hold. Why then is the OLS estimator biased?