Melnick 34 (Mk\,34) is one of the most massive binary systems known and is one of the brightest X-ray point sources in the 30 Doradus region. We investigated the impact of this massive system on the surrounding interstellar medium (ISM) using the optical spectroscopic capabilities of the narrow-field mode (NFM) of the Multi-Unit Spectroscopic Explorer (MUSE). MUSE-NFM spatially resolved the ISM in the vicinity of Mk\,34 with a resolution comparable to that of the HST. The analysis of the N ii \,lambda \,6583 and S ii \,lambda \,6717 emission lines reveals a cone-like structure apparently originating from Mk\,34 and extending southeast. Electron density maps and radial velocity measurements of the ISM lines further support an outflow scenario traced by these emissions. While no clear northwestern counterpart to this outflow was observed, we note increased extinction in that direction, towards the R136 cluster. The ISM material along the projected diagonal of the outflow on both sides of Mk\,34 shows similar properties in terms of the emission line ratios seen in the Baldwin-Phillips-Terlevich diagram. These results are consistent across two observational epochs. Additionally, we examined the residual maps within a 0.5" radius of Mk\,34 after modeling and subtracting the point spread function. The observed variations in the residuals could potentially be linked to Mk\,34's known periodic behavior. However, further observations with appropriate cadence are needed to fully monitor the 155\,day periodicity of Mk\,34's X-ray emissions.