Many industrial machine learning (ML) systems require frequent retraining to keep up-to-date with constantly changing data. This retraining exacerbates a large challenge facing ML systems today: model training is unstable, i.e., small changes in training data can cause significant changes in the model's predictions. In this paper, we work on developing a deeper understanding of this instability, with a focus on how a core building block of modern natural language processing (NLP) pipelines---pre-trained word embeddings---affects the instability of downstream NLP models. We first empirically reveal a tradeoff between stability and memory: increasing the embedding memory 2x can reduce the disagreement in predictions due to small changes in training data by 5% to 37% (relative). To theoretically explain this tradeoff, we introduce a new measure of embedding instability---the eigenspace instability measure---which we prove bounds the disagreement in downstream predictions introduced by the change in word embeddings. Practically, we show that the eigenspace instability measure can be a cost-effective way to choose embedding parameters to minimize instability without training downstream models, outperforming other embedding distance measures and performing competitively with a nearest neighbor-based measure. Finally, we demonstrate that the observed stability-memory tradeoffs extend to other types of embeddings as well, including knowledge graph and contextual word embeddings.