= Solution
The stipulated <DNA> <volume> is $V_D=20000\,\mathrm{nm^3}$ and the spherical <capsid> <volume> is $V_c=4\pi(20\,\mathrm{nm})^3/3=33510\,\mathrm{nm^3}$. Thus
$$
\boxed{\phi=\frac{V_D}{V_c}=0.597\simeq60\%.}
$$
The packed <DNA> occupies a substantial fraction of the available space: the remaining nominal fraction is $40.3\%$. Compared with the paper's stated $0.71$ reference, it reaches about $84\%$ of that value. This indicates dense confinement, but does not prove crystalline order or specify interstrand spacing. <DNA> is a connected elongated <polymer>, rather than a collection of independent hard spheres.
There is also a numerical qualification: the exact densest equal-<sphere packing> fraction is $\pi/(3\sqrt2)\simeq0.74048$, not $0.71$. Using it changes the comparison to about $81\%$ without changing the calculated $0.597$ <volume fraction> or its physical interpretation.
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