Any particular reason why you picked 1924? That date seems to be right on the edge of when space travel and space-based science fiction began to emerge in popular culture. The first movie that really brought "space travel" into the public eye (from a cinematic standpoint) was a film called Woman in the Moon which came out in 1929. By the 1930s, Buck Rogers and Flash Gordon made concepts such as space ships and extraterrestrial aliens much more common place, and of course the broadcast of the radio drama The War of the Worlds happened in 1938.

However, your question was specifically regarding the words "spaceship" and "space- alien", and that wording changes it a bit. From 1924 to 1926 the rocket scientist Konstantin Tsiolkovsky published a series of essays and theoretical formulas that were titled "The Spaceship". I would say that if the term "spaceship" wasn't familiar to most people, they were at least familiar enough with the concept of space-faring rockets to understand what you're referring to.

I think it's likely you'd have to explain it. As /u/Erilaz said, Tsiolkovsky's work doesn't come until 1924, and Google Ngram doesn't show any mentions of the term "spaceship" before 1900, only the briefest mentions between 1900 and 1910, and the term doesn't really take off (pardon the pun) until 1940.

The oldest mention I can find in the archives of The New York Times is this 1931 book review of By Rocket to the Moon: The Story of Hans Hardt's Miraculous Flight, a children's book originally published in Germany.

According to the Online Etymology Dictionary the term "spaceship" dates to 1894 in the novel "A Journey in Other Worlds" by John Jacob Astor. I'm not familiar with the novel so I don't know anything about it (Jules Verne had written about what we would call spaceships now in his science fiction, such as in "From the Earth to the Moon in the 1860s). I couldn't find anything to suggest how popular or well known the book was. I did find out that the author was the richest person to die on the Titanic, so that was interesting if not exactly helpful. http://en.wikipedia.org/wiki/John_Jacob_Astor_IV

Further, according to the Online Etymology Dictionary "alien" being used to mean not of the Earth dates to 1920 which is just before your chosen year of 1924. Once again though it's pretty vague on this one as well.