A hatbox (sometimes spelt hat box) was a box to store hats.
The Fourth Doctor jokingly told George Litefoot that Leela was found floating down the Amazon in a hatbox. (TV: The Talons of Weng-Chiang) Leela later told Ellie Higson that she had lost her luggage in the same manner. (AUDIO: Dead Men's Tales)
Donna Noble had a hatbox prepared on the off chance she would meet the Tenth Doctor again. She did and handed it to him to carry, saying she was ready for the "Planet of the Hats". (TV: Partners in Crime)
According to the Ninth Doctor, the Corsair kept a sub-dimensional void inside a hatbox 23rd century Swindon. (COMIC: Weapons of Past Destruction)