Natural disasters and emergencies pose significant challenges to communities, causing extensive loss of life, economic disruptions, and material damage. As the frequency, intensity, and cost of such events continue to rise, effective disaster management strategies are essential to mitigate their impact. A critical component of disaster preparedness is the planning of logistics depots, and the prepositioning of relief supplies, which play a vital role in ensuring an efficient and organized response. This study addresses the problem of determining the optimal locations for logistics depots and the prepositioning of relief supplies to enhance disaster preparedness. We propose a two-stage robust optimization approach that accounts for uncertainties in demand and travel distances. Our model aims to minimize total costs associated with installation, procurement, transportation, retention, and deprivation. We solve the two-stage model using a column-and-constraint generation algorithm and demonstrate its application through a computational study focused on the Kartal district of Istanbul, Türkiye, considering a potential earthquake scenario.