Cloud services based on content sharing and distribution are most prevalent in today's world, and datacenter (DC) networks supporting such services manage a huge volume of data transfers, in the range of terabytes to zettabytes. Usually, contents are replicated in geographically distributed DCs for high availability and reliability, but in the case of high-impact, large-scale disasters, like weapons of mass destruction attacks, DC networks can suffer massive service disruptions and data loss. To save critical data under such circumstances, contents could be evacuated in response to an upcoming disaster alert from a likely disaster region to a safe location before the disaster occurs and causes serious data damage. Depending on the forecasted disaster scenario, content evacuation can be greatly constrained by limited available network resources and strict deadlines (evacuation times). We propose a rapid-data-evacuation (RDE) heuristic that selects the least-delay paths (considering propagation delays, network bandwidth, and congestion) through an anycast network model, and schedules critical and vulnerable contents for evacuation such that the maximum amount of contents can be evacuated within the evacuation deadline or equivalently, a given amount of contents can be evacuated in minimum time. We compare our RDE heuristic with a nearest-evacuation approach, which evacuates data only to the nearest DC using the shortest path. Our results show that, for typical scenarios considered in this study, compared to nearest evacuation, our rapid-evacuation approach provides about 64% time savings, or equivalently, about 97% more volume of evacuated contents for a given deadline. Since our algorithm is based on a greedy approach, we also present an enhanced RDE algorithm based on the simulated annealing (SA) metaheuristic as a benchmark. We show that our proposed RDE heuristic performs very close to the SA based algorithm and is much faster in computation time. Hence, it is suitable and efficient for rapid evacuation. [ABSTRACT FROM PUBLISHER]