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FaaS (Function-as-a-Service) revolutionized cloud computing by replacing persistent virtual machines with dynamically allocated resources. This shift trades locality and statefulness for a pay-as-you-go model more suited to variable and infrequent workloads. However, the main challenge is to adapt services to the serverless paradigm while meeting functional, performance, and consistency requirements. In this work, we push the boundaries of FaaS computing by designing a serverless variant of ZooKeeper, a centralized coordination service with a safe and wait-free consensus mechanism. We define synchronization primitives to extend the capabilities of scalable cloud storage and outline a set of requirements for efficient computing with serverless. In FaaSKeeper, the first coordination service built on serverless functions and cloud-native services, we explore the limitations of serverless offerings and propose improvements essential for complex and latency-sensitive applications. We share serverless design lessons based on our experiences of implementing a ZooKeeper model deployable to clouds today. FaaSKeeper maintains the same consistency guarantees and interface as ZooKeeper, with a serverless price model that lowers costs up to 110-719x on infrequent workloads.