We present new decentralized storage systems that are resilient to arbitrary failures of up to a half of all servers and can tolerate a computationally unbounded adversary. These are the first such results with space requirements smaller than those of full replication without relying on cryptographic assumptions. We also significantly reduce share sizes for robust secret-sharing schemes with or without an honest dealer, again without cryptographic assumptions. A major ingredient in our systems is an information verification scheme that replaces hashing (for storage systems) or information checking protocols (for secret sharing). Together with a new way of organizing verification information, this allows us to use a simple majority algorithm to identify with high probability all servers whose information hasn't been corrupted.