The Sperm Bank was an operational cryobank aboard the JMC mining spaceship Red Dwarf. It was presumably run by Doc Newton and/or the Medi-Bot, and held the DNA of some - if not all - of the male human members of the crew.
Three million years later aboard Starbug, the last human being Dave Lister received a distress call from his old girlfriend Kristine Kochanski on a wrecked ship. Kochanski claimed that she had also been in stasis, and had twin sons with Lister - Jim and Bexley Lister - which had conceived through use of the sperm bank on Red Dwarf, all without Dave's knowledge.
Lister ordered the others to charge up the Bazookoids and plot a course, but Rimmer and Kryten were quick to point out that Kochanski was still very much dead and that what Lister was seeing was a telepathic hallucination designed to trap Lister by a Psiren. ("Psirens", Series VI)
When the Starbuggers met a real, live version of Kochanski from another dimension, she wanted children. Her boyfriend, a hologram version of Lister, was unable to give her children, and also the Sperm Bank on the other Red Dwarf was apparently not operational anymore. She asked the alive Lister to fill up a tube with his sperm. This would later be used to create a clone of Lister, which he placed in a box under a gravity pool table in the Aigburth Arms in 2155 with the use of the Time Drive, thus revealing that Lister is his own father and completing the time loop of his life. ("Ouroboros", Series VII)
Lister was terrified of eventually ending up as a sperm on the backwards Earth.