Technologically, either a space elevator or a mass driver.

Financially, the best way to fund it would be a special progressive head tax, from $40 to $25,000 per person per year, depending on earnings bracket. I recommend making it a special tax because people will be more willing to fund something if the amount is fixed and for a specific purpose rather than getting funneled into the general fund - they can make the connection between their remittance and what the state is doing with it.