We propose a protocol to encode classical bits in the measurement statistics of many-body Pauli observables, leveraging quantum correlations for a random access code. Measurement contexts built with these observables yield outcomes with intrinsic redundancy, something we exploit by encoding the data into a set of convenient context eigenstates. This allows to randomly access the encoded data with few resources. The eigenstates used are highly entangled and can be generated by a discretely-parametrized quantum circuit of low depth. Applications of this protocol include algorithms requiring large-data storage with only partial retrieval, as is the case of decision trees. Using -qubit states, this Quantum Random Access Code has greater success probability than its classical counterpart for and than previous Quantum Random Access Codes for . Furthermore, for , it can be amplified into a nearly-lossless compression protocol with success probability and compression ratio . The data it can store is equal to Google-Drive server capacity for , and to a brute-force solution for chess (what to do on any board configuration) for .