Inspirations - The Space Museum

This story has never been known as anything other than The Space Museum. That's because it is set in a museum, which is in space. Or at least on another planet. Which is in space.
The fact that the travellers have arrived in a museum seems relevant in the first episode, but matters less as the story progresses.
It is really a conventional revolution / alien invasion plot. A planet - in this case Xeros - has been invaded by aliens - the Moroks - and the Xerons rise up and win their world back.
Now if you've listened to the DVD commentary you'll know that story editor Dennis Spooner rewrote large chunks of this script, which the credited writer - Glyn Jones - was not happy about. Unusually, Spooner actually removed some of the humour. He tended to add jokes.
Talking of jokes, there is an old one amongst fans regarding this story - namely that The Space Museum has three things wrong with it: episodes two, three and four...

The first episode is intriguing, and is certainly the better of the four. For the first time in the series, time travel plays a crucial part. For a series about a time machine, the mechanics of time travel are rarely used dramatically throughout the classic era of the show.
The TARDIS crew, at the end of the previous story, still dressed in their Crusading clothes, suddenly become frozen where they stand as the ship's lights dim. As this story opens, they wake to find themselves standing in their modern outfits. The historical gear is stowed away in the closet. Time has jumped forward. The Doctor shrugs this off - it's what happens when you travel through time - but the companions are obviously baffled. Vicki drops a glass of water, and it suddenly reconstitutes itself and jumps back into her hand. The ship then lands on the planet Xeros, where they see the museum on the scanner. Venturing outside they find it to be a dry, dusty world. The planet name derives from the Greek word for "dry". The travellers find that they are not making any footprints in the dust.
They go to the museum and find that the Morok guards, and later some of the young Xerons, cannot see or hear them. Nor can they hear what the Moroks or Xerons say. Vicki then finds that she can pass her hand through a solid museum display. Eventually the crew find a room in which they see themselves on show as exhibits - along with the TARDIS.
The Doctor realises that they have jumped a time track. They have arrived too early in their own time stream. They must wait for time to catch up with them, and when it does they will be seen and heard - leading to them ultimately being turned into exhibits.

That's all just the first episode, and it's wonderful stuff. We then get the boring and bored Moroks, led by Governor Lobos. Jones got the names from "morons" and "lobotomy". These guys are supposed to be stupid hulking brutes. The Xerons, meanwhile, are all trendy jazz-age teenage boys, dressed in cool black. There's a battle of the generations behind the battle to win back the planet. The hipsters want to kick out the boring trad old dads.
The concept of the teenager was a post-war phenomenon. Before the war, children went to bed one night and woke up the next day as young adults. The war led to a fracturing of family life, with many children growing up without an adult role model - mainly boys not having seen their fathers for many years, assuming they made it back at all. Youngsters had more disposable income once we got into the 1950's, and so lots of people wanted them to spend it on them. Venues opened that catered for younger people - such as the coffee bars - and new music was developed that was geared towards them. Young people themselves began to make the music.
Adults did not know how to react to this new phenomenon, and so there was much social conflict between them and the older generation. Church and State, and the Sunday papers, predicted the breakdown of society. Every time the oldies pushed, the teenagers just pushed back.
Observing the zombies today staring at their mobile phones, the oldies might have had a point...

One of the museum exhibits is a Dalek, of the type seen in their very first story. This gives Hartnell the chance to impersonate one, when he hides in the empty casing. Ian thinks it unlikely they will encounter them again - a production in-joke as the next story will see them return, and the closing sequence trails this.
If there is one Greek legend that Doctor Who has touched upon more than any other over the decades, it is that of the Minotaur. Ian decides to dismantle Barbara's cardigan and use the wool to guide them through the labyrinth of the museum.
The word "Museum" derives from a place sacred to the Muses. They were the nine daughters of Zeus. A couple of their names will be familiar to Doctor Who fans, as they were used for characters - e.g. Erato and Thalia (a big green blob and a Member of the High Council of Time Lords). They represented various artistic forms - different forms of poetry, dance, and so forth. A Macedonian king had nine daughters and decided to name them after the Muses. He thought them more gifted than the goddesses, and they were all turned into magpies. Fickle lot, the Greek pantheon.
There was a museum in Alexandria in the 3rd Century BC, but the modern concept of a museum begins with the Ashmolean in Oxford, which opened its doors in 1683. It was designed to house the bequest to the university from Elias Ashmole. Rome's Capitoline Museum was the first art collection to be owned by the public (1471, when Pope Sixtus IV donated a collection).

Back to the time mechanics. the idea of people coming across alternative versions of themselves is an old one. There's a 1963 Twilight Zone episode starring Jack Klugman called "Death Ship", where a group of astronauts arrive on a planet and find a crashed spaceship identical to their own - containing their own dead bodies. Space: 1999 did something similar much later.
Much of this Doctor Who story is taken up with the Doctor and friends debating predestination and free will. What do they have to do to stop themselves ending up as exhibits? Each action they take is debated - is it taking them closer to the glass cases, or away from them? The Doctor is sure that what they have seen is one possible future, and need not come to pass. Their interactions with the Xerons and the Moroks will have had an impact - throwing many variables into the mix. As it is, it is Vicki - the one who doesn't see the point in worrying about consequences - who helps with the rebellion, breaking into the armoury, and so leading to the downfall of Lobos and his men.
The explanation for the time track jump turns out to be a prosaic one. Jones had it that it was the Morok equipment which caused the problem, whereas Spooner makes it yet another TARDIS fault - another stuck switch (as with Edge of Destruction the year before).
Next time - a works outing for the Daleks. Ian dad-dances, Barbara loses another cardigan, Vicki discovers that the Beatles played classical music, and the Doctor beats himself up...