We know that the Project had a satellite in orbit (at least one, maybe more). It may have had the ability to boost small payloads into orbit after the war to replace other satellites for various purposes. It had a great interest in the Houston Space Center, ostensibly to control the existing satellite, but more likely for a much broader set of circumstances and capabilities, since a good old dish and vehicle mobile system could probably do the chore of managing one satellite and its data pretty well.
With fusion power and cryostorage space travel to even near stars is a reasonably doable prospect, once you can get everything you need into orbit. The catch here is getting material into orbit. The Project could use some sort of Project Orion type booster, since they appear to have pure fusion devices there will be a lot less fallout and residual radiation from such a vehicle.
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