It is proved that $EX^2 < \infty$ is necessary for a very mild form of the bootstrap of the mean to work a.s. and that $X$ must be in the domain of attraction of the normal law if a.s. is weakened to "in probability."
"Necessary Conditions for the Bootstrap of the Mean." Ann. Statist. 17 (2) 684 - 691, June, 1989. https://doi.org/10.1214/aos/1176347134