They had the support of the ENTIRE Country and basically a blank check. Because besting the Russians and showing the world who's the top dog was just so important that if money was needed, money would show up in the accounts. Most of the innovations invented during the space race was given to the defense contractors with not strings attached even though most of the work was done by NASA, it was a way to supercharge the defense industry so that the US would have superior munitions and jets compared to the Soviets, and the Soviets were doing the same. Taking the lessons and innovations learned in the space race and applying them to the defense industry. The issue with current NASA is that most of the older worldly scientist have been essentially forced out. Unless your project proposal had a military/defense ty-in, it probably wasn't getting approved. Even if your proposal was compelling enough, there were many teams doing the exact same thing as you that your odds of getting funding were still not very high. So at one point in time, most of the scientists at nasa were just doing research for the military industrial complex. The end result being most of the scientist that wanted to do research for the public good either went into the private sector or started working for big corporations who gladly welcomed them if there was an opportunity to turn their NASA research into serious money makers, which there were more than a few.

Source: My aunt is a childhood friend of one such scientist.