Vessel structural support system for laterally and vertically supporting a vessel, such as a nuclear steam generator having an exterior bottom surface and a side surface thereon. The system includes a bracket connected to the bottom surface. A support column is pivotally connected to the bracket for vertically supporting the steam generator. The system also includes a base pad assembly connected pivotally to the support column for supporting the support column and the steam generator. The base pad assembly, which is capable of being brought to a level position by turning leveling nuts, is anchored to a floor. The system further includes a male key member attached to the side surface of the steam generator and a female stop member attached to an adjacent wall. The male key member and the female stop member coact to laterally support the steam generator. Moreover, the system includes a snubber assembly connected to the side surface of the steam generator and also attached to the adjacent wall for dampening lateral movement of the steam generator. In addition, the system includes a restraining member of "flat" attached to the side surface of the steam generator and a bumper attached to the adjacent wall. The flat and the bumper coact to further laterally support the steam generator.