Introduced by Emek and Wattenhofer (PODC 2013), the \emph{stone age (SA)} model provides an abstraction for network algorithms distributed over randomized finite state machines. This model, designed to resemble the dynamics of biological processes in cellular networks, assumes a weak communication scheme that is built upon the nodes' ability to sense their vicinity in an asynchronous manner. Recent works demonstrate that the weak computation and communication capabilities of the SA model suffice for efficient solutions to some core tasks in distributed computing, but they do so under the (somewhat less realistic) assumption of fault free computations. In this paper, we initiate the study of \emph{self-stabilizing} SA algorithms that are guaranteed to recover from any combination of transient faults. Specifically, we develop efficient self-stabilizing SA algorithms for the \emph{leader election} and \emph{maximal independent set} tasks in bounded diameter graphs subject to an asynchronous scheduler. These algorithms rely on a novel efficient self-stabilizing \emph{asynchronous unison (AU)} algorithm that is "thin" in terms of its state space: the number of states used by the AU algorithm is linear in the graph's diameter bound, irrespective of the number of nodes.