[Usagi (OP)]

I've read about this project a lot and even read that book by Dyson, but it seems to me no one ever made not even half serious calculation how much this would cost. What I have in mind is just a robotic probe (no people on board) which is cheaper to fly to some of the nearby stars. Not only to to explore other star system but outer reaches of our own, which could be very exciting. What lies beyond Kuiper belt and then the mystical Oort cloud.

So it would be made of few big parts (from bottom up):
1. Giant shield made of steel with diameter of 400 meters.
2. 600 atom bombs and mechanism that stores them and then shoots them trough the hole in the mentioned steel shield
3. scientific instruments like telescopes, cameras
4. nuclear reactor to harness electricity for many decades
5. computer with really smart software
6. on the top a small shield made of steel for the debris ahead

$$$$

1. Shield with 400 m diameter with height let's say 25 m would be 62800m3 volume. Weight of steel is 7850 kg/m3 = 492.980.000 kilograms of steel. As I see price of steel seems to be $50 per tonne which is 492.980 t × 50 = 2,4 billion dollars.

2. 600 atom bombs from what I see, on wiki page on Orion, a bomb could be made even little bit less then $1 million so let's say it's $600 million. With mechanism which I don't know how much but let's say $50 million.

3. Telescopes, cameras, transmitters etc. I don't know how much but let's say $10 billion.

4. Now nuclear reactor is the biggest problem because you couldn't use ones we have now since they can't last up to 100 years, not even couple of decades. You would have to use so called traveling-wave reactor (TWR) which is fission reactor that uses nuclear waste. It's a small thing that uses small amount of nuclear waste and can go sealed for something like 50 years and then other one can be switched on for next 50 years. This is a thing that doesn't exist yet, but if it did it doesn't seem that it would be that expensive. And developing it would find use in every day life.

5. Supercomputer, again I don't know how much to price it so let's say $20 million.

6. top shield again few dozen million dollars, let's say $30 million.

So in the end I get something like 2,6 billion dollars. And most of that is without transport and other construction work. So even if it doubles it it's "only" 6 billion dollars.

What do you think?

[Atomic-S]

You probably have to lift it off the earth by conventional propulsion due to concerns about radiation pollution in the atmosphere.

[chiralSPO]

It costs $10-20 per gram to get a payload into Low Earth orbit. At this rate, it would require almost 7.4 trillion dollars to get only the shield into orbit. (not counting the cost of developing a launch system capable of lifting such a monstrous payload.)

One might be able to assemble the shield in space, but it is unclear whether this would actually save any money, as no one has ever tried anything of this nature.

[Usagi (OP)]

You probably have to lift it off the earth by conventional propulsion due to concerns about radiation pollution in the atmosphere.

Actually no. Even in Dysons book it is stated Orion could be launched over the North Pole where the magnetic field lines go straight out, and use very unsymmetrical bombs so you can be sure the debris is going outwards and not inwards. To soften this even further neutron bombs could be used since they produce neutrons without fission, and neutrons you could easily absorb. They calculated if the above method was used the given radiation dispersed around the world would be high enough to kill "only" 1 person.
And reading Wiki article on Orion they mention that there are atom bombs that are almost 100% fusion reaction which means almost no fallout.

Also I didn't add the price very good at the end. So we have column of billions:
2,4 billion
10 billion

Then millions:
650 million
20 million
30 million

So it's $12,4 billion 700 million (most of that is without transport and other construction work).

Oh but imagine the adventure!! It would be the excitement that humanity has not felt since New World was discovered. And also the presence of building such a spaceship would be motivation to many millions people to turn to physics and other sciences.