Anna and I actually did a Fermi estimate of the fuel for the Apollo 11 mission over dinner last week, and we were off by a factor of two. Some of the available inputs:

• A crude estimate of the potential energy of mass lifted from Earth's surface to a distance of many times its radius
• The heights reached by jet aircraft using fuel amounting to only a very small portion of their mass
• A crude estimate of the energy content of gasoline (one approximation is to energy content of food, and/or the energy output of humans), with adjustment for the need to carry oxygen into space
• Images of rockets launching, which show that the fuel tanks are much bigger than payload, but not thousands or millions of times bigger
• Knowledge of the price of consumer gasoline, or the price of oil
• The existence of science fiction writers with physics backgrounds, SpaceX, the L-5 societies, and other groups seriously pushing for advancements to slash cost-to-orbit
• Rough knowledge of NASA's budget, either directly or by bounding it relative to known US budget items
• Knowledge of the enormous cost of producing military aircraft and naval vessels, which can be in the hundreds of millions to billions of dollars
• The existence of an ecology of NASA contractors condemned for their enormous costs (these would be trivial if fuel was the major cost)

Estimate how many people you would have needed to employ and what their salaries might be, estimate the cost of materials...