Consider $\mathcal{B}$ the Borel $\sigma$-algebra of Lebesgue measurable subsets of $[0,1]$. We then define a pseudometric $d(A,B) = \mu(A \Delta B)$ on $\mathcal{B}$ where $A \Delta B = A \setminus B \cup B \setminus A$ is the symmetric difference of sets. Ignoring sets of zero measure, we have that $d$ is indeed a metric. We wish to show that $\mathcal{B}$ is not sequentially compact in its metric.
Write $x = d_0.d_1d_2\dots$ for the binary expansion of $x \in [0,1]$. Consider $E_i = \{x = d_0.d_1d_2\dots\ :\ d_i = 1\}$. We compute $E_i \Delta E_j = \{x : d_i \neq d_j\}$. If we think probabilistically, where the digits $d_1$ and $d_j$ represent independent coin tosses, we get: $\mu(E_i \Delta E_j) = 1/2$ for $i \neq j$. Thus $d(E_i,E_j)$ is constant for $i \neq j$ and hence $\{E_i\}$ can have no convergent subsequences