We still do, but we're no longer in an existential war with a major world power so we tend to avoid blank-check crash-development programs.

Let me take a moment here to debunk the concept that the Apollo program was such a good thing. I love the Apollo program, I think it was a marvelous thing and a remarkable achievement. It remains deeply inspirational. But it was flawed. Deeply, deeply flawed. It was a war-time stunt more than anything. It expended vast amounts of resources for very little practical benefit, even on a basic "blue sky" research level. At the end of Apollo we had spent somewhere around a hundred billion dollars (in today's money) and yet we were little closer to the moon than before. It still cost us a tremendous billion dollars a passenger or more to put a man on the moon for at most a few days. Because Apollo was a race. We didn't build infrastructure, we didn't build systems that made spaceflight easier. And then afterward when we attempted to build infrastructure with the Shuttle and the ISS we screwed up so badly that it ended up being even more expensive to get to LEO than it had been to get to the Moon with Apollo. We spent a quarter of a trillion dollars building the ISS and it hasn't provided as much benefit toward exploring or colonizing the solar system as you'd expect spending that much money should.

The idea that we need big government projects to solve "big" problems is part of the problem. What we need is merely basic research and sound engineering.

As it turns out, we are starting to solve big problems. It's happening right under our noses. In space exploration SpaceX is rapidly leading the way to a new renaissance of commercial spaceflight and exploration (both manned and unmanned). They have already built the most cost effective launch vehicle in existence and are in the midst of building a reusable launch vehicle, a heavy lift launcher, and a super heavy lift launcher that will dwarf the Saturn V. All of which should facilitate the creation of a great deal of industrial infrastructure in Earth orbit which will reduce the barrier to exploration of the solar system.

Let's look at other projects which could be considered "big problems" or engineering marvels. In many cases it's been a matter of engineering marvels becoming commonplace. Enormous bridges, tunnels, transoceanic cables, etc. all of these things are no longer marvels they are just ordinary engineering. The channel tunnel was a tremendous engineering marvel, but today it has become just an accepted matter of engineering, and many similarly audacious tunnel projects have been completed since, to relatively minor fanfare. The same goes for bridges. And today there is always at least one transoceanic fiber optical cable in the midst of being laid down.

Meanwhile, let's look at the big problems of our age, such as cancer, disease, poverty. In the last half century several countries have pulled themselves out of poverty and become developed countries (South Korea, Taiwan, etc.) And several other countries are on the same path. A smaller percentage of the world is in poverty today than throughout history. Several types of cancer today have 80-90% or higher average survival rates. Major diseases (smallpox, rinderpest) have been eradicated. And major advances against diseases such as malaria and AIDS are being made by the day. In some cases due to extensive amounts of resources being brought to bear and in other cases merely due to the ordinary workings of trade and industry.

There are many big problems that remain unsolved, of course, but we should not wallow in the mistaken notion that we are any less capable of tackling such problems or that we need to mobilize heaven and Earth in order to make progress.