[abridged] The ALESS survey has followed-up a sample of 122 sub-millimeter sources in the Extended Chandra Deep Field South at 870um with ALMA, allowing to pinpoint the positions of sub-millimeter galaxies (SMGs) to 0.3 and to find their precise counterparts at different wavelengths. This enabled the first compilation of the multi-wavelength spectral energy distributions (SEDs) of a statistically reliable survey of SMGs. In this paper, we present a new calibration of the MAGPHYS modelling code that is optimized to fit these UV-to-radio SEDs of z>1 star-forming galaxies using an energy balance technique to connect the emission from stellar populations, dust attenuation and dust emission in a physically consistent way. We derive statistically and physically robust estimates of the photometric redshifts and physical parameters for the ALESS SMGs. We find that they have a median stellar mass $M_ast=(8.9pm0.1)times10^{10} M_odot$, SFR$=280pm70 M_odot$/yr, overall V-band dust attenuation $A_V=1.9pm0.2$ mag, dust mass $M_rm{dust}=(5.6pm1.0)times10^8 M_odot$, and average dust temperature Tdust~40 K. The average intrinsic SED of the ALESS SMGs resembles that of local ULIRGs in the IR range, but the stellar emission of our average SMG is brighter and bluer, indicating lower dust attenuation, possibly because they are more extended. We explore how the average SEDs vary with different parameters, and we provide a new set of SMG templates. To put the ALESS SMGs into context, we compare their stellar masses and SFRs with those of less actively star-forming galaxies at the same redshifts. At z~2, about half of the SMGs lie above the star-forming main sequence, while half are at the high-mass end of the sequence. At higher redshifts (z~3.5), the SMGs tend to have higher SFR and Mstar, but the fraction of SMGs that lie significantly above the main sequence decreases to less than a third.
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