How widespread was the notion that space travel was impossible prior to the successful launch of Sputnik 1?

Clarke later apologized for misquoting Wooley on the subject of spaceflight. The passage in question came from a radio interview, so there isn't a solid paper trail, but there's no doubt that summing up Wooley's position as "spaceflight is utter bilge" is totally unfair. Wooley was talking about the popular idea that interplanetary flight would soon be widespread, and said (according to a 1995 letter to the New Scientist): "All this talk about space travel is utter bilge, really. It would cost as much as a major war just to put a man on the moon."

Clarke acknowledged that Wooley was right on the money if you substitute "minor war" for "major war." In any case, he was far more correct than Clarke at the time, who thought that going to the Moon could be accomplished for a few million dollars.

I have serious trouble believing anyone in the physics community actually believed rockets wouldn't work in a vacuum. The principle that allows them to operate there is Newton's Third Law, which by the 20th century had been around quite a while.

True, random people off the street may have believed that, but random people off the street today often have trouble figuring out why jumping in an airplane doesn't ram you against the back wall. So personally I don't think opinions of the subject-matter ignorant should count that much. The problem comes in when you rely on such people for your funding.

I do think you are slightly misreading the quotes in your linked article¹, but you came by it honestly because you were meant to misunderstand in exactly that way. A cynic might argue this is journalists glossing over the minor fact they caused the problem.

There are two parts here. First off:

Too often the word "impossible" is used by people who lack the imagination or even the knowledge to understand what is being attempted.

IOW, we are talking my proverbial random people off the street here, not people who actually understand physics.

Then there's this (quoting Goddard):

The subject of projection from the earth, and especially a mention of the moon, must still be avoided in dignified scientific and engineering circles.

Nowhere does it say "because those circles think its impossible". The article implies that by placing those quotes in proximity (so bad on it), but I don't think that's what Goddard was saying. He was referring to public perception making it tough to get things like funding and publication. He had trouble getting taken seriously because the general public associated his rockets with cheap Sci-Fi.

In the early and mid 20th century Science Fiction was largely cheap schlocky paperbacks, and had only slightly more cachet than Romance novels. It was looked down upon by pretty much anyone "serious"²

Gooddard mentioned the moon in an aside in a large report on his rocketry research. Some ignoramus (obviously not an actual physicist) wrote an anonomous Op-Ed to the New York Times, ridiculing the entire research based on this tiny bit, with indeed the claim that somehow Newton's Third Law doesn't work in a vacuum.³

After the rocket quits our air and really starts on its longer journey, its flight would be neither accelerated nor maintained by the explosion of the charges it then might have left. To claim that it would be is to deny a fundamental law of dynamics, and only Dr. Einstein and his chosen dozen, so few and fit, are licensed to do that.

This article had no small effect on history, as the public ridicule caused others in the USA to avoid Goddard and his rocketry research. For what its worth (almost nothing), the Times did eventually publish a retraction ... in 1969.

Further investigation and experimentation have confirmed the findings of Isaac Newton in the 17th Century and it is now definitely established that a rocket can function in a vacuum as well as in an atmosphere. The Times regrets the error.

(emphasis mine)

^{1 - On the first page of your link. I actually couldn't find any mention of space travel on the other articles under it, but that could be my old eyes.}

^{2 - I can remember in the mid 80's being specifically told by an English prof I could read and write a report on any book I wanted, but it shouldn't be Romance or Sci-Fi because those weren't "serious literature". I got ticked off and wrote a 20 page mini-dissertation on the themes in Dune}

^{3 - Notice this also including a reference to the Urban Legend that almost nobody can understand Relativity. As an indirect result of this letter, it was the Nazis who ended up picking up his research and making the real advancements in this field. Really I can't say anything bad enough about the anonymous coward who penned this.}

According to books like Profiles of the Future by Arthur C. Clark and Rockets, Missiles and Space Travel by Wily Ley, many scientists and engineers misunderstood the physics enough to claim that space travel was impossible or impractical in reaction to discussions of the theories of Goddard, Oberth, and other space travel pioneers in the early 20th century.

For example, Clarke mentioned an article denying the practicality of space travel by a Richard V. D. W. Wooley. Wooley later became Astronomer Royal in the UK in 1956 and told reporters that space travel was "utter bilge". After the first Sputnik the next year a famous cartoon showed him being shown around a space center and being told "...and then, Dr. Wooley, the satellite will penetrate utter bilge."

On the other hand, a newspaper-like publication distributed to American schoolchildren during the International Geophysical Year mentioned that the USA and the USSR planned to launch satellites during it, so the launch of Sputnik should not have been too surprising to kids who read it.

In the early and mid 20th century Science Fiction was largely cheap schlocky paperbacks, and had only slightly more cachet than Romance novels. It was looked down upon by pretty much anyone serious2

A number of serious persons, such as scientists and engineers, wrote science fiction stories themselves in the early and mid 20th century.

During the early and mid 20th century science fiction was mostly Buck Rogers and Flash Gordon type comic strips, stories and novels published in the mainstream "slick" magazines, occasional more or less mainstream science fiction hardback novels, and stories and novels published in the science fiction genre "pulp" magazines. Mass market genre science fiction paperback novels, story collections, and anthologies did not begin to be published until after World War II -- the earliest in my collection have book copyright dates in the 1950s and so could be considered to be from the later mid 20th century.