Lycosa tarantula was the scientific name for a species of spider that originated from Spain on Earth. Some were known to be kept as pets. At some point, a number of tarantulas of this type were taken to the planet Titus IV.
Miles O'Brien kept a tarantula named "Christina" as a pet in the early 2360s while he served aboard the USS Enterprise-D. Though Chief O'Brien was once afraid of spiders, he took her in as a pet after almost stepping on her on Titus IV. He once showed Christina to Lieutenant Barclay, who wasn't particularly fond of it. (TNG: "Realm of Fear")