In many engineering applications, data samples are expensive to get and limited in number. In such a difficult context, this paper shows how classification based on Reproducing Kernel Hilbert Space (RKHS) can be used in conjunction with Extreme Value Theory (EVT) to estimate extreme multivariate quantiles and small probabilities of failure. For estimating extreme multivariate quantiles, RKHS one-class classification makes it possible to map vector-valued data onto $\RR$, so as to estimate a high quantile of a univariate distribution by means of EVT. In order to estimate small probabilities of failure we basically apply multivariate EVT, however EVT is hampered by the fact that many samples may be needed before observing a single tail event. By means of a new method again based on RKHS classification, we can partially solve this problem and increase the proportion of tail events in the samples collected.