We present an optimal emulation of a server based regular read/write storage in a synchronous round-free message-passing system that is subject to mobile Byzantine failures and prove that the problem is impossible to solve in asynchronous settings. In a system with $n$ servers implementing a regular register, our construction tolerates faults (or attacks) that can be abstracted by agents that are moved (in an arbitrary and unforeseen manner) by a computationally unbounded adversary from a server to another in order to deviate the server's computation. When a server is infected by an adversarial agent, it behaves arbitrarily until the adversary decides to "move" the agent to another server. We investigate the case where the movements of the mobile Byzantine agents are decided by the adversary and are completely decoupled from the message communication delay. Our emulation spans two awareness models: servers with and without self-diagnose mechanism. In the first case servers are aware that the mobile Byzantine agent has left and hence they can stop running the protocol until they recover a correct state while in the second case, servers are not aware of their faulty state and continue to run the protocol using an incorrect local state. Our results, proven optimal with respect to the threshold of the tolerated mobile Byzantine faults, are significantly different from the round-based synchronous models. Another interesting side result of our study is that, contrary to the round-based synchronous consensus implementation for systems prone to mobile Byzantine faults, our storage emulation does not rely on the necessity of a core of correct processes all along the computation. That is, every server in the system can be compromised by the mobile Byzantine agents at some point in the computation. This leads to another interesting conclusion: storage is easier than consensus in synchronous settings, when the system is hit by mobile Byzantine failures.