This paper presents Elevator-LIO, a LiDAR-inertial odometry framework designed to achieve continuous robot localization during elevator travel, thereby supporting cross-floor robotic tasks. To address the state-estimation problem in non-inertial frames, Elevator-LIO establishes a decoupled state-estimation model that separately models the robot motion relative to the elevator and the elevator motion itself, and embeds it into a mode-dependent iterated error-state Kalman filter framework. This framework degenerates to conventional LIO estimation in ordinary indoor environments, while enabling the propagation and constrained update of elevator-related states in elevator non-inertial environments, thereby achieving continuous and stable localization. An elevator mode manager detects elevator entry and exit events using LiDAR ranging statistics and estimated states, and introduces event-triggered zero-velocity and zero-acceleration updates when the elevator stops to suppress accumulated vertical drift. In addition, this paper adopts an adaptive voxel downsampling strategy to maintain a stable number of effective points under significant environmental scale changes. We conduct extensive experiments on 20 real-world sequences containing 79 elevator rides, including practical challenges such as large-scale spaces, long vertical travel, dynamic pedestrian interference, and mirror reflections. The results show that Elevator-LIO maintains continuous localization accuracy in all sequences, with terminal height error below 1 cm in 17 sequences. In contrast, existing representative localization systems perform poorly on these elevator sequences. Tests on the Hilti 2022/2023 datasets further show that the proposed method remains competitive in standard indoor scenarios. The project page is available at https://xiaofan4122.github.io/Elevator_LIO_Page/.