Using the Density Matrix Renormalization Group technique we study the effect of spin-orbit coupling on a three-orbital Hubbard model in the $(t_{2g})^{4}$ sector and in one dimension. Fixing the Hund coupling to a robust value compatible with some multiorbital materials, we present the phase diagram varying the Hubbard $U$ and spin-orbit coupling $lambda$, at zero temperature. Our results are shown to be qualitatively similar to those recently reported using the Dynamical Mean Field Theory in higher dimensions, providing a robust basis to approximate many-body techniques. Among many results, we observe an interesting transition from an orbital-selective Mott phase to an excitonic insulator with increasing $lambda$ at intermediate $U$. In the strong $U$ coupling limit, we find a non-magnetic insulator with an effective angular momentum $langle(textbf{J}^{eff})^{2}rangle e 0$ near the excitonic phase, smoothly connected to the $langle(textbf{J}^{eff})^{2}rangle = 0$ regime. We also provide a list of quasi-one dimensional materials where the physics discussed in this publication could be realized.
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