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)