= Solution
The barred notation denotes the antichiral <supersymmetric covariant derivative>. Write barred spinor indices as dotted indices, and use the stated plus-sign convention with <left Grassmann derivatives>:
$$
\bar D_{\dot\alpha}=\partial_{\bar\theta^{\dot\alpha}}+i\theta^\beta\sigma^\mu_{\beta\dot\alpha}\partial_\mu,\qquad
D_\alpha=\partial_{\theta^\alpha}+i\sigma^\mu_{\alpha\dot\beta}\bar\theta^{\dot\beta}\partial_\mu.
$$
Their <supercovariant derivative algebra with left derivatives> gives $\{\bar D_{\dot\alpha},\bar D_{\dot\beta}\}=0$ and $\{D_\alpha,\bar D_{\dot\beta}\}=2i\sigma^\mu_{\alpha\dot\beta}\partial_\mu$. The constraint $\boxed{\bar D_{\dot\alpha}\Phi=0}$ defines a <chiral superfield>, also called an <L-superfield>.
To solve the constraint, put $y^\mu=x^\mu+i\theta\sigma^\mu\bar\theta$. For a <left Grassmann derivative>, $\partial_{\bar\theta^{\dot\alpha}}(\theta\sigma^\mu\bar\theta)=-\theta^\beta\sigma^\mu_{\beta\dot\alpha}$. Therefore $\bar D_{\dot\alpha}y^\mu=0$ and $\bar D_{\dot\alpha}\theta^\beta=0$, so $\bar D$ becomes ordinary differentiation with respect to $\bar\theta$ at fixed $y$. Its general solution is the <chiral-superfield component expansion>
$$
\boxed{\Phi=\Phi(y,\theta)=A(y)+\sqrt2\theta^\alpha\chi_\alpha(y)+\theta\theta F(y).}
$$
Here $A$ is a complex scalar, $\chi$ a <Weyl spinor>, and $F$ a complex <auxiliary field>. Before imposing field <equations of motion> this is four real bosonic and four real fermionic components. Chirality means independence of $\bar\theta$ at fixed $y$, rather than at fixed $x$. The differential <supercharges> $Q_\alpha=\partial_{\theta^\alpha}-i\sigma^\mu_{\alpha\dot\beta}\bar\theta^{\dot\beta}\partial_\mu$ and $\bar Q_{\dot\alpha}=\partial_{\bar\theta^{\dot\alpha}}-i\theta^\beta\sigma^\mu_{\beta\dot\alpha}\partial_\mu$ anticommute with both covariant derivatives, so a <supersymmetry> transformation preserves the chiral constraint.
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