We obtained three epochs of simultaneous Hubble Space Telescope (HST)/Wide Field Camera 3 and Chandra observations of the ultraluminous X-ray source (ULX) NGC 5408 X-1. The counterpart of the X-ray source is seen in all HST filters, from the UV through the near-IR (NIR), and for the first time, we resolve the optical nebula around the ULX. We identified a small OB association near the ULX that may be the birthplace of the system. The stellar association is young, ~5 Myr, contains massive stars up to 40 M &sun;, and is thus similar to associations seen near other ULXs, albeit younger. The UV/optical/NIR spectral energy distribution (SED) of the ULX counterpart is consistent with that of a B0I supergiant star. We are also able to fit the whole SED from the X-rays to the NIR with an irradiated disk model. The three epochs of data show only marginal variability and thus, we cannot firmly conclude on the nature of the optical emission.