IRAS 08339+6517 is a luminous infrared and Ly$α$-emitting starburst galaxy that possesses a dwarf companion object at a projected distance of 56 kpc. An \ion{H}{i} tidal tail has recently been detected between both galaxies, suggesting that about 70% of the neutral gas has been ejected from them.We present deep broad-band optical images, together with narrow band H$α$ CCD images, and optical intermediate-resolution spectroscopy of both galaxies. The images reveal interaction features between both systems and strong H$α$ emission in the inner part of IRAS 08339+6517. The chemical composition of the ionized gas of the galaxies is rather similar. The analysis of their kinematics also indicates interaction features and reveals an object that could be a candidate tidal dwarf galaxy or a remnant of an earlier merger. Our data suggest that the \ion{H}{i} tail has been mainly formed from material stripped from the main galaxy. We find weak spectral features that could be attributed to the presence of Wolf--Rayet stars in this starburst galaxy and estimate an age of the most recent burst of around 4 -- 6 Myr. A more evolved underlying stellar population, with a minimal age between 100 -- 200 Myr, is also detected and fits an exponential intensity profile. A model which combines 85% young and 15% old populations can explain both the spectral energy distribution and the \ion{H}{i} Balmer and \ion{He}{i} absorption lines presented in our spectrum. The star formation rate of the galaxy is consistently derived using several calibrations, giving a value of $∼$9.5 \Mo yr$^{-1}$. IRAS 08339+6517 does satisfy the criteria of a luminous compact blue galaxy, rare objects in the local universe but common at high redshifts, being a very interesting target for detailed studies of galaxy evolution and formation. ; Comment: 23 pages, 8 tables, 18 figures. Accepted for publication in Astronomy & Astrophysics