I would argue that the Concorde was from the beginning economically unfeasible and instead served as a prestige technology for the British and French governments. Even the name Concorde, indicates that the two governments working together to produce the aircraft seemed the most important aspect of it.
Both British and France at the time had completely nationalized airlines and almost all their major aviation companies as well. Americans were going to the moon so Britain and France were looking around for a prestige technology of their own. Bigger and faster airplanes had been the rage for twenty years so the Concorde seemed like a cool thing to do. The Soviets came to the same conclusion.
In the US, civilian airline development was still wholly private so when when Boeing , Douglas et crunched the numbers they saw little value in supersonic transport with it's non-linear cost. US supersonic transport never made it off the ground, but in Europe, the politically driven Concorde did because the people making the decisions weren't paying the bills.
As noted by others, the Concorde was essentially a military aircraft, produced by political compromise with the primary goal of expressing the technical powers of the governments involved. It was more an art project like building the Pyramids than it was a piece of mobile transportation infrastructure.
Edit: (May 14, 2014)
Some sources on the economics of the Concorde:
The Atlantic August 1977:
Already Britain, and France spent £1.46 billion \$2.3 billion to reach this (and a London political economist has recently argued that
the true cost is roughly three times this amount)...
...In May 1976, Professor David Henderson, newly appointed professor of political economy at University College, London, argued
that the government's figure of £1.46 billion shared between Britain
and France was a drastic underestimate. It had been reached by adding
the yearly expenditure on the project at the current prices. If these
were adjusted to 1975 prices, and interest charges of 10 percent
added, then the cost of Concorde was not £1.46 bilion but £4.26
billion (\$6.82 billion at the present exchange rate of \$1.60).
That would be $22 billion US (2014 dollars.) Just to get the thing into the air.
They built 26 of the aircraft, 20 of which that actually flew paid flights. So, around 1.1 billion per plane, roughly the cost of a B2 stealth bomber. I can't find any hard numbers on operating cost but with 1.1 billion per plane sunk capitial cost, it's clear it would never operate at a total profit.
I believe the plane only flew for 20+ years because the British government essentially gave it away, eating the billions of in development and deployment cost and leaving the plane to just have to cover operating cost. I'm not clear that it even really did that.
Considering how much money was thrown at the problem, not particularly elegant or interesting.
In rereading the parent, I realized I didn't make my answer explicit to the original question. He wanted to know what it would take to bring a Concorde like plane back.
- My answer was really just saying, "the Concorde itself had no economic foundation so firstly, you'd have to find an economic niche for SST."
- There was never an economic demand for the Concorde's speed of travel and there doesn't seem to be one now, either.
- Technologically, you'd probably need something hyper radical like a nuclear powered air spike so the plane wouldn't burn tons of fuel and could fly over land masses without sonic booms or destroying smaller aircraft. (Seriously, it would have to be that advanced.)
- In our current luddite era, it would be saner to think about inventing anti-gravity drives than think that the political regulators would let that type of radical technology fly, even if it existed.