We present the first results from the X-SHOOTER Lyman-$\alpha$ survey at $z=2$ (XLS-$z2$). XLS-$z2$ is a deep spectroscopic survey of 35 Lyman-$\alpha$ emitters (LAEs) utilising $\approx90$ hours of exposure time with VLT/X-SHOOTER and covers rest-frame Ly$\alpha$ to H$\alpha$ emission with R$\approx4000$. We present the sample selection, the observations and the data reduction. Systemic redshifts are measured from rest-frame optical lines for 33/35 sources. In the stacked spectrum, our LAEs are characterised by an interstellar medium with little dust, a low metallicity and a high ionisation state. The ionising sources are young hot stars that power strong emission-lines in the optical and high ionisation lines in the UV. The LAEs exhibit clumpy UV morphologies and have outflowing kinematics with blue-shifted SiII absorption, a broad [OIII] component and a red-skewed Ly$\alpha$ line. Typically 30 % of the Ly$\alpha$ photons escape, of which one quarter on the blue side of the systemic velocity. A fraction of Ly$\alpha$ photons escapes directly at the systemic suggesting clear channels enabling a $\approx10$ % escape of ionising photons, consistent with an inference based on MgII. A combination of a low effective HI column density, a low dust content and young star-burst determine whether a star forming galaxy is observed as a LAE. The first is possibly related to outflows and/or a fortunate viewing angle, while we find that the latter two in LAEs are typical for their stellar mass of 10$^9$ M$_{\odot}$.