Although memories are largely thought to be encoded in the brain via modifications to synaptic weights, technical and conceptual issues with this view (e.g., the large disparity between the time scale of synaptic turnover and long-term human memories) lead some to hypothesize that intracellular processes like gene regulation and post-translational modifications support complementary memory storage mechanisms. While this possibility is appealing, it lacks the mature theoretical apparatus associated with canonical theories of neural network memory storage and recall. To ameliorate this issue, we introduce a theoretical framework for studying intracellular memory, which applies an information-theoretic perspective to biophysically realistic models of intracellular processes, including signaling cascades, gene regulation, and epigenetic processes. We identify two strong biophysical constraints on the design of intracellular memory devices: omnipresent and non-negligible intrinsic noise, and characteristic time scales on which noise affects the integrity of device components like proteins. By studying the extent to which intracellular memory devices can reliably store a single bit of information on long time scales, we identify three design principles that support performance despite these biophysical constraints. The first, which we call “local stability”, generalizes an idea due to Crick that devices ought to have an error-correction mechanism to combat intrinsic noise. The second, “redundancy”, suggests that the same information ought to be encoded in multiple ways. The third, “hierarchy”, suggests that intracellular memory devices can overcome time scale constraints by appropriately coupling devices that operate on different time scales. This allows devices to both learn quickly and maintain memories on long time scales, and suggests that intracellular memory systems ought to exploit many types of processes (e.g., protein modifications, gene regulation, epigenetics), since each operates on a somewhat different time scale. These principles sharpen the concept of 'memory molecules' and suggest signatures of cellular memory.