Step 1: What we want.
Packing efficiency is the share of the unit cell space actually filled by the atoms, written as a percentage.
Step 2: Facts about BCC.
A body-centred cubic cell has $2$ atoms in it. The atoms touch along the body diagonal, giving $4r = \sqrt{3}\,a$, so $a = \dfrac{4r}{\sqrt{3}}$.
Step 3: Volume of the cell.
$a^3 = \left(\dfrac{4r}{\sqrt{3}}\right)^3 = \dfrac{64r^3}{3\sqrt{3}}$.
Step 4: Volume of the atoms.
The two atoms fill $2 \times \dfrac{4}{3}\pi r^3 = \dfrac{8}{3}\pi r^3$.
Step 5: Take the ratio.
Efficiency $= \dfrac{\frac{8}{3}\pi r^3}{\frac{64r^3}{3\sqrt{3}}}\times 100 = \dfrac{\sqrt{3}\,\pi}{8}\times 100$. Using $\pi \approx 3.14$ and $\sqrt{3}\approx 1.732$, this is about $68\%$.
Step 6: Final choice.
The packing efficiency of BCC is $68\%$, which is option 4.
\[ \boxed{68\%} \]