Ignoring the effect of atmospheric refraction, distance to the horizon from an observer close to the Earth's surface is about:$ d \approx 3.57\sqrt{h} $

Given $h = 6 ft = 1.83 m \rightarrow d = \approx 5km$

Given that the horizon is $5km$, when the sun sets again, it passes this distance of the horizon.,

We know that this must be equal to the angle of: $2\pi$ $\large \frac{6 \times 240 }{24 \times 60 }$

But this we know = $\large\frac{d}{R} \rightarrow $$R = \large \frac{5km}{2\pi}$$ = 796km$, which we know cannot be the radius of the earth.

Therefore, we can infer that the time interval the sun takes to pass the horizon in this case is too long and cannot be 6 minutes, but should be in the order of magnitude of a few seconds.

We know that the radius of the earth is 6350km, so working backwards, we can see that this must be: $\approx \large\frac{24\times60\times60\times5km}{6350km\times2\pi}$$=10.83s$ and couldn't have been $6mins$ (Or this must be another smaller earth-like planet) :)