There is described a multiple rack apparatus (1) for accommodating biological product containers (2) unloaded from a storage (3) for the preservation of the same interfaced with a laboratory automation system (4), comprising an odd number of said racks (7) which can be housed within a platform (6) having a number of locations (12, 120) which is equal to said odd number plus one. Said racks (7) can horizontally or vertically slide from one location (12, 120) to another of said platform (6), taking advantage of the presence of a temporarily empty location (12, 120), said racks (7) further occupying in turn an operating location (120) which, being vertical with respect to an outlet tube (5) of said storage (3), accommodates said biological product containers (2) unloaded from said storage (3).