We present ORACLE, the first hierarchical deep-learning model for real-time, context-aware classification of transient and variable astrophysical phenomena. ORACLE is a recurrent neural network with Gated Recurrent Units (GRUs), and has been trained using a custom hierarchical cross-entropy loss function to provide high-confidence classifications along an observationally-driven taxonomy with as little as a single photometric observation. Contextual information for each object, including host galaxy photometric redshift, offset, ellipticity and brightness, is concatenated to the light curve embedding and used to make a final prediction. Training on $\sim$0.5M events from the Extended LSST Astronomical Time-Series Classification Challenge, we achieve a top-level (Transient vs Variable) macro-averaged precision of 0.96 using only 1 day of photometric observations after the first detection in addition to contextual information, for each event; this increases to $>$0.99 once 64 days of the light curve has been obtained, and 0.83 at 1024 days after first detection for 19-way classification (including supernova sub-types, active galactic nuclei, variable stars, microlensing events, and kilonovae). We also compare ORACLE with other state-of-the-art classifiers and report comparable performance for the 19-way classification task, in addition to delivering accurate top-level classifications much earlier. The code and model weights used in this work are publicly available at our associated GitHub repository (https://github.com/uiucsn/ELAsTiCC-Classification).