First, I believe that this nation should commit itself to achieving the goal, before this decade is out, of landing a man on the moon and returning him safely to the Earth.
20 days before the speech, the US had only just sent its first American on a suborbital flight into space. How did the US have enough confidence in their ability to land on the moon despite no human doing more than orbiting the earth at that time?
Was the success measured in just the attempt at a moon landing, even if it didn't turn out to be physically possible for humans to reach or land on the moon? Did we know enough through science and the technology available to know for certain it was an achievable goal at that time, or was Kennedy hoping we could just figure it out in time?
It wasn't Kennedy's estimate; it was originally from James Webb, administrator of NASA. In March 1961 he sent a request for a budget increase with the goal of reaching the Moon before the end of the decade. A "after 1970" estimate had previously been floated in a 1959 long range plan and a 1960 report estimated the landing at 1975, so this was clearly a bump in ambition.
A month later, JFK later sent a letter to Johnson (Chairman of the Space Council) asking
Do we have a chance of beating the Soviets by putting a laboratory in space, or by a trip around the moon, or by a rocket to land on the moon, or by a rocket to go to the moon and back with a man. Is there any other space program which promises dramatic results in which we could win? ... How much additional would it cost?
The questions made their way to von Braun who wrote directly to Johnson later that same month.
He estimated that the US
did not have a chance beating the Soviets to a laboratory in space; they had rocket capability of doing so the same year whereas von Braun estimated the Saturn C-1 to be available in 1964.
had a "sporting chance" of being the first for a radio transmitter station on the moon, which could happen in early 1962.
had a "sporting chance" (again) of being the first to send a crew around the moon, estimating dates of 1965/1966, but he thought the Soviets could make it by 1962 or 1963 if they short-cutted on safety.
finally, the US had an "excellent" chance of being the first to land a crew on the moon, because the performance of rockets required was much higher than the Soviet capability (von Braun estimated by a factor of 10). So while the Americans did not have such a rocket at that time, the Soviets didn't either, and he estimated with "an all-out crash program" it could happen by 1967/1968.
These estimates, combined with Webb's, combined with some consultation with military and industry leaders, all made it into a report from Johnson to Kennedy.
Since Kennedy wanted to find some way to "win" the space race, this led to the "before the end of the decade" deadline in the famous speech before Congress.
To a real extent, the obstacle here was money. This was astronomical compared to what had come before; for example, January 1960's budget request from NASA was a mere $915 million. Johnson's memo to JFK indicates NASA would need $1 billion extra a year over their current budget and that while NASA was working at full capacity, "This work can be speeded up through firm decisions to go ahead faster if accompanied by additional funds needed for the acceleration." The naysayers (famously, including former president Eisenhower) tended to focus on the financial aspect, not the technical infeasibility.
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Logsdon, J. M. (2010). John F. Kennedy and the Race to the Moon. Springer.
Rosholt, R. L. (1966). An Administrative History of NASA 1958-1963. NASA SP-4101. NASSP, 4101.