We present a sample of $\sim 1000$ emission line galaxies at $z=0.4-4.7$ from the $\sim0.7$deg$^2$ High-$z$ Emission Line Survey (HiZELS) in the Bo\"otes field identified with a suite of six narrow-band filters at $\approx 0.4-2.1$ $\mu$m. These galaxies have been selected on their Ly$\alpha$ (73), [OII] (285), H$\beta$/[OIII] (387) or H$\alpha$ (362) emission-line, and have been classified with multi-wavelength photometry, multiple narrow-band (e.g. [OII]-H$\alpha$) detections and spectroscopy. In this paper, we present the observations, selection and catalogs of emitters, the general properties of the sample and the first results. We derive luminosity functions (LFs) for Ly$\alpha$, [OII], H$\beta$/[OIII] and H$\alpha$ and confirm a strong luminosity evolution from $z\sim0.4$ to $\sim 5$, in good agreement with previous results obtained in other fields like COSMOS and UDS. We explore the properties of dual line-emitters, most notably [OII]-H$\alpha$ at $z=1.47$. The observed [OII]/H$\alpha$ ratio increases from 0.40$\pm0.01$ at $z=0.1$ to 0.52$\pm0.05$ at $z=1.47$, which we attribute to either decreasing dust attenuation with redshift, or due to fiber-measurements in the local Universe which only measure the central kpc regions. At the bright end, we find that both the H$\alpha$ and Ly$\alpha$ LFs at $z\approx2.2$ deviate significantly from a Schechter form, becoming a power-law. We show that this is fully driven by an increasing X-ray/AGN fraction with line-luminosity, reaching $\approx 100$ % at line-luminosities $\gtrsim3\times10^{44}$ erg s$^{-1}$.