We present the implementation of a hybrid continuum-atomistic model for including the effects of a surrounding electrolyte in large-scale density functional theory (DFT) calculations within the Order-N Electronic Total Energy Package (ONETEP) linear-scaling DFT code, which allows the simulation of large complex systems such as electrochemical interfaces. The model represents the electrolyte ions as a scalar field and the solvent as a polarizable dielectric continuum, both surrounding the quantum solute. The overall energy expression is a grand canonical functional incorporating the electron kinetic and exchange-correlation energies, the total electrostatic energy, entropy, and chemical potentials of the surrounding electrolyte, osmotic pressure, and the effects of cavitation, dispersion, and repulsion. The DFT calculation is performed fully self-consistently in the electrolyte model, allowing the quantum-mechanical system and the surrounding continuum environment to interact and mutually polarize. A bespoke highly parallel multigrid Poisson-Boltzmann solver library, DL-MG, deals with the electrostatic problem, solving a generalized Poisson-Boltzmann equation. Our model supports open boundary conditions, which allows the treatment of molecules, entire biomolecules, or larger nanoparticle assemblies in the electrolyte. We have also implemented the model for periodic boundary conditions, allowing the treatment of extended systems such as electrode surfaces in contact with the electrolyte. A key feature of the model is the use of solute size and solvation-shell-aware accessibility functions that prevent the unphysical accumulation of electrolyte charge near the quantum solute boundary. The model has a small number of parameters - here we demonstrate their calibration against experimental mean activity coefficients. We also present an exemplar simulation of an 1634-atom model of the interface between a graphite anode and LiPF 6 electrolyte in an ethylene carbonate solvent. We compare the cases where Li atoms are intercalated at opposite edges of the graphite slab and in solution, demonstrating a potential application of the model in simulations of fundamental processes in Li-ion batteries.