Context: The blazar Markarian 421 is one of the brightest TeV gamma-ray sources of the northern sky. From December 2007 until June 2008 it was intensively observed in the very high energy (VHE, E > 100 GeV) band by the single-dish Major Atmospheric Gamma-ray Imaging Cherenkov telescope (MAGIC-I). Aims: We aimed to measure the physical parameters of the emitting region of the blazar jet during active states. Methods: We performed a dense monitoring of the source in VHE with MAGIC-I, and also collected complementary data in soft X-rays and optical-UV bands; then, we modeled the spectral energy distributions (SED) derived from simultaneous multi-wavelength data within the synchrotron self--compton (SSC) framework. Results: The source showed intense and prolonged gamma-ray activity during the whole period, with integral fluxes (E > 200 GeV) seldom below the level of the Crab Nebula, and up to 3.6 times this value. Eight datasets of simultaneous optical-UV (KVA, Swift/UVOT), soft X-ray (Swift/XRT) and MAGIC-I VHE data were obtained during different outburst phases. The data constrain the physical parameters of the jet, once the spectral energy distributions obtained are interpreted within the framework of a single-zone SSC leptonic model. Conclusions: The main outcome of the study is that within the homogeneous model high Doppler factors (40 <= delta <= 80) are needed to reproduce the observed SED; but this model cannot explain the observed short time-scale variability, while it can be argued that inhomogeneous models could allow for less extreme Doppler factors, more intense magnetic fields and shorter electron cooling times compatible with hour or sub-hour scale variability.
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