Due to the rising frequency of climate change and extreme weather events, the modern power system is more dependent on flexible resources. Most current studies focus on the planning of electricity resources within a single timescale, overlooking the coordination needs for multi-timescale flexibility resources under extreme weather conditions. In response to this issue, this paper quantifies the impact of climate change and extreme weather on the multi-timescale flexibility demand of the power system through meteorological data analysis, and introduces a multi-timescale energy storage coordination planning model to more fully evaluate and optimize energy storage system designs. The study first clarifies the mechanisms of climate change and extreme weather on the power system, quantifies the variation of wind and solar resources under non-disastrous extreme weather, and further explores the impact of different power expansion scenarios on energy storage demand. By comparing the energy storage demand parameters under different scenarios, the paper reaches the following key conclusions: changes in climate averages can reduce the energy storage demand in regions with high heating load shares to some extent; extreme low-output events lead to greater fluctuations in the power generation capacity of new energy sources, significantly increasing the demand for long-duration energy storage, especially for long-duration storage during the daytime; as the installed capacity of new energy sources increases, the demand for long-duration energy storage becomes particularly prominent, playing a key role in ensuring grid stability and responding to extreme weather events.