Cloud object storage such as AWS S3 is cost-effective and highly elastic but relatively slow, while high-performance cloud storage such as AWS ElastiCache is expensive and provides limited elasticity. We present a new cloud storage service called ServerlessMemory, which stores data using the memory of serverless functions. ServerlessMemory employs a sliding-window-based memory management strategy inspired by the garbage collection mechanisms used in the programming language to effectively segregate hot/cold data and provides fine-grained elasticity, good performance, and a pay-per-access cost model with extremely low cost. We then design and implement InfiniStore, a persistent and elastic cloud storage system, which seamlessly couples the function-based ServerlessMemory layer with a persistent, inexpensive cloud object store layer. InfiniStore enables durability despite function failures using a fast parallel recovery scheme built on the auto-scaling functionality of a FaaS (Function-as-a-Service) platform. We evaluate InfiniStore extensively using both microbenchmarking and two real-world applications. Results show that InfiniStore has more performance benefits for objects larger than 10 MB compared to AWS ElastiCache and Anna, and InfiniStore achieves 26.25% and 97.24% tenant-side cost reduction compared to InfiniCache and ElastiCache, respectively.