I subscribed to NASA Tech Briefs for several years, probably a decade in all from the mid-70s through mid-90s. I sometimes was pleasantly surprised by clever ideas, but I do not recall any truly game-changing invention. Mostly, they produced a lot of incremental improvements to technology they inherited (bought).
The Amateur Radio Relay League's OSCAR I was lauched on December 12, 1961.
The satellite was built, quite literally, in the basements and garages of the Project OSCAR team. It was the first satellite to be ejected as a secondary payload from a primary launch vehicle and then enter a separate orbit. This was accomplished using a very high technology and thermally balanced ejection system: a $1.15 spring from Sears. The total out-of-pocket cost (not including material donations) of OSCAR I: only $68.
AT&T's Telstar was launched seven momths later, on July 10, 1962.
Robert Goddard's initial explorations with liquid fueled rockets were funded by the Smithsonian (5 yerars $5000) and then Guggenheim Foundation ($100,000 over four years), large sums in those days, but modest by all measures. In terms of gold the first $5000 would be like half a million now. The Guggenheim grant (after devaluation of the dollar to $32 per ounce) would be like $6.25 million - over four years. It wold be sizable, but not unreasonable. Long ago (1971?), Murray Rothbard had an essay called "The Government Airplane." He contrasted Samuel P. Langley with the Wright Brothers. It is a story now retold in many libertarian webspaces.
To me the overarching narrative is Bastiat's broken window, alluded to by Stephen above. We see the glass being replaced. We do not see the losses. Perhaps a helpful analogy is to the settlement of North America and the founding of the United States. In strict libertarian terms, the early adventures by Columbus, Hudson, and others who sold their enterprises to the crown were all losses. But, eventually, it led to profitable enterprises never imagined. The Spanish sought gold. The French sought furs. The English just wanted to life their own lives.
BTW - I think that Mars is doable, just not by humans as we define ourselves now. We are still only able to support 1,000 to 4,000 in the Antarctic and they are not self-sufficient. Give it time.