We investigate, by molecular dynamics simulation, the behaviour of discotic particles in a solvent of Lennard-Jones spheres. When chromonic disc -sphere interactions are imposed on these systems, three regimes of self-assembly are observed. At moderate temperatures, numerous short threads of discs develop, but these threads remain isolated from one another. Quenching to low temperatures, alternatively, causes all of the discs to floc into a single extended aggregate which typically comprises several distinct sections and contains numerous packing defects. For a narrow temperature range between these regimes, however, defect-free chiral fibres are found to freely self-assemble. The spontaneous chirality of these fibres results from frustration between the hexagonal packing and interdigitation of neighbouring threads, the pitch being set by the particle shape. This demonstration of aggregate-wide chirality emerging owing to packing alone is pertinent to many biological and synthetic hierarchically self-assembling systems.