The connection and the metric of a manifold are technically two independent quantities. If we allow both of those to be independent, then making a flat manifold with torsion is fairly trivial, for instance

\begin{eqnarray} ds^2 &=& -dt^2 + dx^idx_i\\ {\Gamma^\alpha}_{\mu\nu} &=& g^{\alpha\beta} \varepsilon_{\beta\mu\nu} \end{eqnarray}

This is one of the simplest connection with a torsion. it can easily be checked to have torsion, via

\begin{equation} {T^\alpha}_{\mu\nu} = {\Gamma^\alpha}_{[\mu\nu]} = g^{\alpha\beta} \varepsilon_{\beta\mu\nu} \end{equation}

with also (for what follows)

\begin{eqnarray} {T}_{\gamma\mu\nu} &=& g_{\alpha\gamma} g^{\alpha\beta} \varepsilon_{\beta\mu\nu}\\ &=& \varepsilon_{\gamma\mu\nu} \end{eqnarray}

and the contorsion tensor

\begin{eqnarray} K_{\alpha\beta\gamma} &=& \frac{1}{2} (T_{\alpha\beta\gamma} - T_{\beta\gamma\alpha} + T_{\gamma\alpha\beta})\\ &=& \frac{1}{2} T_{\alpha\beta\gamma} \end{eqnarray}

You can check fairly easily that despite the arbitrariness of our connection choice, it has torsion but it is still a metric connection :

\begin{eqnarray} \nabla_\alpha g_{\mu\nu} &=& \partial_\alpha g_{\mu\nu} - {\Gamma^\beta}_{\mu\alpha} g_{\beta\nu} - {\Gamma^\beta}_{\nu\alpha} g_{\mu\beta}\\ &=& - g^{\beta\gamma} \varepsilon_{\gamma\mu\alpha} g_{\beta\nu} - g^{\beta\gamma} \varepsilon_{\gamma\nu\alpha} g_{\mu\beta}\\ &=& - \varepsilon_{\nu\mu\alpha} - \varepsilon_{\mu\nu\alpha}\\ &=& 0 \end{eqnarray}

So that we have a connection that has torsion but is nethertheless metric, so that it still fits within the framework of the Einstein-Cartan theory, if we so wish.

There is in fact a whole theory based on that notion called teleparallel gravity, which contains a flat connection with torsion, and it is an entirely acceptable theory of gravity.

The connection and the metric of a manifold are technically two independent quantities. If we allow both of those to be independent, then making a flat manifold with torsion is fairly trivial, for instance

\begin{eqnarray} ds^2 &=& -dt^2 + dx^idx_i\\ {\Gamma^\alpha}_{\mu\nu} &=& g^{\alpha\beta} \varepsilon_{\beta\mu\nu} \end{eqnarray}

This is one of the simplest connection with a torsion. it can easily be checked to have torsion, via

\begin{equation} {T^\alpha}_{\mu\nu} = {\Gamma^\alpha}_{[\mu\nu]} = g^{\alpha\beta} \varepsilon_{\beta\mu\nu} \end{equation}

with also (for what follows)

\begin{eqnarray} {T}_{\gamma\mu\nu} &=& g_{\alpha\gamma} g^{\alpha\beta} \varepsilon_{\beta\mu\nu}\\ &=& \varepsilon_{\gamma\mu\nu} \end{eqnarray}

and the contorsion tensor

\begin{eqnarray} K_{\alpha\beta\gamma} &=& \frac{1}{2} (T_{\alpha\beta\gamma} - T_{\beta\gamma\alpha} + T_{\gamma\alpha\beta})\\ &=& \frac{1}{2} T_{\alpha\beta\gamma} \end{eqnarray}

You can check fairly easily that despite the arbitrariness of our connection choice, it has torsion but it is still a metric connection :

\begin{eqnarray} \nabla_\alpha g_{\mu\nu} &=& \partial_\alpha g_{\mu\nu} - {\Gamma^\beta}_{\mu\alpha} g_{\beta\nu} - {\Gamma^\beta}_{\nu\alpha} g_{\mu\beta}\\ &=& - g^{\beta\gamma} \varepsilon_{\gamma\mu\alpha} g_{\beta\nu} - g^{\beta\gamma} \varepsilon_{\gamma\nu\alpha} g_{\mu\beta}\\ &=& - \varepsilon_{\nu\mu\alpha} - \varepsilon_{\mu\nu\alpha}\\ &=& 0 \end{eqnarray}

So that we have a connection that has torsion but is nethertheless metric, so that it still fits within the framework of the Einstein-Cartan theory, if we so wish. You can also check that this is flat via

\begin{eqnarray} {R^\alpha}_{\mu\nu\sigma} &=& 2 {\Gamma^\alpha}_{[\nu | \lambda} {\Gamma^\lambda}_{\sigma| \mu]}\\ &=& g^{\alpha\beta} g^{\lambda\beta} (\varepsilon_{\beta\nu\lambda} \varepsilon_{\gamma\sigma\mu} - \varepsilon_{\beta\mu\lambda} \varepsilon_{\gamma\sigma\nu})\\ &=& 0 \end{eqnarray}

As we are trying to contract a tensor symmetric in $\lambda \beta$ and one antisymmetric in it.

There is in fact a whole theory based on that notion called teleparallel gravity, which contains a flat connection with torsion, and it is an entirely acceptable theory of gravity.