Aims: We present a study of the very high-energy (VHE; E> 100 GeV) γ-ray emission of the blazar PKS 1424+240 observed with the MAGIC telescopes. The primary aim of this paper is the multiwavelength spectral characterization and modeling of this blazar, which is made particularly interesting by the recent discovery of a lower limit of its redshift of z ≥ 0.6 and makes it a promising candidate to be the most distant VHE source. Methods: The source has been observed with the MAGIC telescopes in VHE γ rays for a total observation time of ~33.6 h from 2009 to 2011. A detailed analysis of its γ-ray spectrum and time evolution has been carried out. Moreover, we have collected and analyzed simultaneous and quasi-simultaneous multiwavelength data. Results: The source was marginally detected in VHE γ rays during 2009 and 2010, and later, the detection was confirmed during an optical outburst in 2011. The combined significance of the stacked sample is ~7.2σ. The differential spectra measured during the different campaigns can be described by steep power laws with the indices ranging from 3.5 ± 1.2 to 5.0 ± 1.7. The MAGIC spectra corrected for the absorption due to the extragalactic background light connect smoothly, within systematic errors, with the mean spectrum in 2009-2011 observed at lower energies by the Fermi-LAT. The absorption-corrected MAGIC spectrum is flat with no apparent turn down up to 400 GeV. The multiwavelength light curve shows increasing flux in radio and optical bands that could point to a common origin from the same region of the jet. The large separation between the two peaks of the constructed non-simultaneous spectral energy distribution also requires an extremely high Doppler factor if an one zone synchrotron self-Compton model is applied. We find that a two-component synchrotron self-Compton model describes the spectral energy distribution of the source well, if the source is located at z ~ 0.6. Appendix is available in electronic form at http://www.aanda.org