In the smallest capillaries, or in tubes with diameter D ≲ 8 μm, flowing red blood cells are well known to organize into single-file trains, with each cell deformed into an approximately static bullet-like shape. Detailed high-fidelity simulations are used to investigate flow in a model blood vessel with diameter slightly larger than this: D = 11.3 μm. In this case, the cells deviate from this single-file arrangement, deforming continuously and significantly. At the higher shear rates simulated (mean velocity divided by diameter U/D ≳ 50s−1), the effective tube viscosity is shear-rate insensitive with μeff/μplasma = 1.21. This matches well with the value μeff/μplasma = 1.19 predicted for the same 30% cell volume fraction by an established empirical fit of high-shear-rate in vitro experimental data. At lower shear rates, the effective viscosity increases, reaching μeff/μplasma ≈ 1.65 at the lowest shear rate simulated (U/D ≈ 3.7s−1). Because of the continuous deformations, the cell-interior viscosity is potentially important for vessels of this size. However, most results for simulations with cell interior viscosity five times that of the plasma (λ = 5), which is thought to be close to physiological conditions, closely match results for cases with λ = 1. The cell-free layer that forms along the vessel walls thickens from 0.3 μm for U/D = 3.7s−1 up to 1.2 μm for U/D ≳ 100s−1, in reasonable agreement with reported experimental results. The thickness of this cell-free layer is the key factor governing the overall flow resistance, and this in turn is shown to follow a trend expected for lubrication lift forces for shear rates between U/D ≈ 8s−1 and U/D ≈ 100s−1. Only in this same range are the cells near the vessel wall on average inclined relative to the wall, as might be expected for a lubrication mechanism. Metrics are developed to quantify the kinematics of this dense cellular flow in terms of the well-known tank-treading and tumbling behaviours often observed for isolated cells in shear flows. One notable effect of λ = 5 versus λ = 1 is that it suppresses treading rotation rates by a factor of about 2. The treading rate is found to scale with the velocity difference across the cell-rich core and is thus significantly slower than the overall shear rate in the flow, which is presumably why the flow is otherwise insensitive to λ. The cells in all cases also have a similarly slow mean tumbling motion, which is insensitive to cell-interior viscosity and decreases monotonically with increasing U/D.