Imagine that one day humanity decides to disassemble the planet and convert it into space colonies with a much greater combined internal surface area than the Earth's prior surface. One possible method of doing this would be to build numerous space elevators: nanotube fiber ropes extending from the equator to a counterweight in geosynchronous orbit. An advanced network of space elevator could carry almost arbitrarily large loads upwards using armies of robotic climbers. Disassembling the entire planet could take a while, though.
The gravitational potential energy of anything in geosynchronous orbit, relative to the Earth's surface, is about 50 MJ (15 kWh) of energy per kilogram. The Earth contains about 6 × 1024 kg of mass, which would necessitate 1.2 × 1032 J of energy to send into GEO from the surface if the gravity were constant. The gravity would not be constant, however: after a significant amount of material is removed from the planet, its gravity would decrease considerably. As a very rough estimate, let's say this effect lowers the energy requirements to about half of what it would be if gravity remained constant at 1 g. We also ignore the complex effects of gravitational interactions between massive colonies in orbit and the energy costs for further dispersal within the Earth-Moon system.
The final estimated energy cost, 6 × 1031 J, is very large, but not necessary out of the reach of a advanced solar civilization. As Arthur C. Clarke said, "Any sufficiently advanced technology is indistinguishable from magic." This value is "only" about a hundred billion times greater than humanity's global energy consumption in 2004. Humanity's power generation and consumption has increased exponentially since the Industrial Revolution. It seems probable that at some point in the distant future we will reach such huge populations and energy generation capacity (solar and nuclear) that disassembling the Earth could become feasible, if it were desired.
Consider using solar power as an energy source for disassembling the Earth. Power could be gathered using an astronomically-sized network solar panels orbiting inside the orbit of Mercury, beaming energy back to Earth using a 50% efficient network of relay stations. The total solar flux is about 4 × 1026 watts. Imagine a giant network of solar panels so large that they absorb a full 1% of the solar flux. Being very thin, they would not take up that much matter, and could be constructed using materials from the asteroid belt.
Assuming 50% efficient solar panels absorbing 1% of the solar flux and sending power back to Earth with 50% efficiency, enough energy to disassemble the Earth could be supplied in just six days.
Of course, the practical matters of building the robots and space elevators and miners to extract all the Earth's material and send it into orbit are imposing. However, if humanity continues existing for many millions of years, we'll have plenty of time to try. Calculations show that enough power is available from the Sun to try it, given sufficiently advanced robotics. Whether disassembling the Earth is really possible or not, we'll just have to wait and see. Less than a hundred years ago, many eminent scientists and rocket experts believed that traveling to the Moon would be physically impossible.
