Many allocation problems in multiagent systems rely on agents specifying cardinal preferences. However, allocation mechanisms can be sensitive to small perturbations in cardinal preferences, thus causing agents who make ``small" or ``innocuous" mistakes while reporting their preferences to experience a large change in their utility for the final outcome. To address this, we introduce a notion of algorithmic stability and study it in the context of fair and efficient allocations of indivisible goods among two agents. We show that it is impossible to achieve exact stability along with even a weak notion of fairness and even approximate efficiency. As a result, we propose two relaxations to stability, namely, approximate-stability and weak-approximate-stability, and show how existing algorithms in the fair division literature that guarantee fair and efficient outcomes perform poorly with respect to these relaxations. This leads us to explore the possibility of designing new algorithms that are more stable. Towards this end, we present a general characterization result for pairwise maximin share allocations, and in turn use it to design an algorithm that is approximately-stable and guarantees a pairwise maximin share and Pareto optimal allocation for two agents. Finally, we present a simple framework that can be used to modify existing fair and efficient algorithms in order to ensure that they also achieve weak-approximate-stability.