We use observational data from Supernovae (SNIa) Pantheon sample, as well as from direct measurements of the Hubble parameter from the cosmic chronometers (CC) sample, in order to extract constraints on the scenario of Barrow holographic dark energy. The latter is a holographic dark energy model based on the recently proposed Barrow entropy, which arises from the modification of the black-hole surface due to quantum-gravitational effects. We first consider the case where the new deformation exponent $\Delta$ is the sole model parameter, and we show that although the standard value $\Delta=0$, which corresponds to zero deformation, lies within the 1$\sigma$ region, a deviation is favored. In the case where we let both $\Delta$ and the second model parameter to be free we find that a deviation from standard holographic dark energy is preferred. Additionally, applying the Akaike, Bayesian and Deviance Information Criteria, we conclude that the one-parameter model is statistically compatible with $\Lambda$CDM paradigm, and preferred comparing to the two-parameter one. Finally, concerning the present value of the Hubble parameter we find that it is close to the Planck value.