Where can I safely store my stuff?
If I store things in a chest in a room I have rented, will it safely stay there like it would in a house I own or will it vanish after a few days?
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Items in any chest that doesn't belong to you can vanish.
Most containers in the game are subject to reset, that is, they clear out all the contents after a certain period of time and are reset to their initial state. Most containers reset after about a week of game time, however, examining the contents of the container causes the countdown to start over. A container could be made to never reset simply by opening it once every 5-6 days or so. Any item a player has left in a container that has reset is lost. Wiki
(Just on a side note, I read a few times that chests had reseted earlier than the 7 days the wiki states as the interval.)