High-resolution spectroscopy and speckle interferometry reveal the young star HD 86588 as a quadruple system with a three-tier hierarchy. The 0.″3 resolved binary A,B with an estimated period around 300 years contains the 8-yr pair Aa,Abc (also potentially resolvable), where Ab,Ac is a double-lined binary with equal components, for which we compute the spectroscopic orbit. Despite the short period of 2.4058 days, the orbit of Ab,Ac is eccentric (e = 0.086 ± 0.003). It has a large inclination, but there are no eclipses; only a 4.4 mmag light modulation apparently caused by star spots on the components of this binary is detected with Evryscope. Assuming a moderate extinction of A V = 0.5 mag and a parallax of 5.2 mas, we find that the stars are on or close to the main sequence (age >10 Myr) and their masses are from 1 to 1.3 solar. We measure the strength of the lithium line in the visual secondary B which, together with rotation, suggests that the system is younger than 150 Myr. This object is located behind the extension of the Chamaeleon I dark cloud (which explains extinction and interstellar sodium absorption), but apparently does not belong to it. We propose a scenario where the inner orbit has recently acquired its high eccentricity through dynamical interaction with the outer two components; it is now undergoing rapid tidal circularization on a timescale of ∼1 Myr. Alternatively, the eccentricity could be excited quasi-stationary by the outer component Aa.