We use deep Very Large Telescope (VLT) optical and near-infrared spectroscopy and deep Spitzer/IRAC imaging to examine the properties of two of the most luminous Lyα emitters at z= 5.7. The continuum redward of the Lyα line is clearly detected in both objects, thus facilitating a relatively accurate measurement (10–20 per cent uncertainties) of the observed rest-frame equivalent widths, which are around 160 Å for both objects. Through detailed modelling of the profile of the Lyα line with a 3D Monte Carlo radiative transfer code, we estimate the intrinsic rest-frame equivalent width of Lyα and find values that are around 300 Å, which is at the upper end of the range allowed for very young, moderately metal-poor star-forming galaxies. However, the uncertainties are large and values as high as 700 Å are permitted by the data. Both Lyα emitters are detected at 3.6 m in deep images taken with the Spitzer Space Telescope. We use these measurements, the measurement of the continuum redward of Lyα and other photometry to constrain the spectral energy distributions of these very luminous Lyα emitters and to compare them with three similar Lyα emitters from the literature. The contribution from nebular emission is included in our models: excluding it results in significantly higher masses. Four of the five Lyα emitters have masses of the order of ∼109 M⊙ and fairly high specific star formation rates (≳10–100 Gyr−1). While our two Lyα emitters appear similar in terms of the observed Lyα rest-frame equivalent width, they are quite distinct from each other in terms of age, mass and star formation history. Evidence for dust is found in all objects, and emission from nebular lines often makes a dominant contribution to the rest-frame 3.6 m flux. Rich in emission lines, these objects are prime targets for the next generation of extremely large telescopes, the James Webb Space Telescope (JWST) and the Atacama Large Millimeter Array (ALMA).