The lift takes them up to the model city where the Mechonoids talk nonsense in a very hard to understand accent - one or the other might have helped make them seem alien, but both together just makes them annoying - and Ian, Barbara and Vicki mock them for it:
""Enter, enter, zero, stop?" What does it mean?"
"It means "Enter, enter, zero, stop.""
Don't worry though, the episode gets very good at this point, because now they meet Steven Taylor...
... and Hi Fi...
Why is Hi Fi not included on the list of official Doctor Who Companions? Me and my friends think he should be.
Steven tells them about the planet Mechanus and how they are all prisoners of the Mechonoids. Steven has been trying to escape for two years, but has not managed it by himself. So this city is the Mechanus equivalent of Colditz castle.
The Doctor and Ian immediately start looking for ways to escape. They go up to the roof where the Mechonoids don't go. The Daleks come to fight the Mechonoids, while the TARDIS crew escape by climbing down a cable.
The Doctor uses the device he has been carrying around since last episode, and it sets the place on fire. Steven runs into the smoke to rescue Hi Fi. The Doctor, Ian, Barbara and Vicki get to the bottom of the cable and away, but there is no sign of Steven or Hi Fi - oh noes! This keeps the suspense level high.
There are exciting Dalek vs Mechonoid scenes with 1960s Batman-style explosions overlaid to make it look even more dramatic. This serves as the climax to the story, with the Daleks and Mechonoids wiping each other out.
At the Dalek time machine, Ian trolls the others by pretending to be a Dalek. Really there are no surviving Daleks so they decide to steal the time machine.
We see Steven and Hi Fi are safe and in the jungle, looking for the Doctor. This is the last we see of them in this episode so we are kept in suspense as to whether or not they found their way safely to the TARDIS. Fortunately I have previously seen the following episode, The Watcher, so I know they do - otherwise it is likely that Steven Taylor would not have been in any more stories, and that would be terrible because he is great!
The Doctor doesn't want Ian and Barbara to risk using the Dalek time machine to get home in case they end up in Spain, or in space, or even in Space-Spain (it's Terry Nation, I wouldn't put it past him, would you?) by accident. They argue, with callbacks to the very first story when Ian and Barbara first met the Doctor:
"I've tried for two years to get you both home!"
"Well you haven't been very successful, have you?"
"How dare you, young man? How dare you, sir! I didn't even invite you into the ship in the first place. You both thrust yourselves upon me!"
It is Vicki who persuades the Doctor to let them try. The Dalek time machine takes Ian and Barbara to 1965 and then blows up. There is a photo montage of Ian and Barbara in London, including them in front of a police telephone box that is not the TARDIS. This shows they have a happy ending after all their adventures.
The Doctor and Vicki see them on the space-time television from the beginning of this story, so they know they are both down and safe. The Doctor is sad and misses them. The Chase thus ends with a bittersweet moment.
"Come along my dear, it's time we were off."
The Chase is not the most consistent of Terry Nation's many great Doctor Who stories, being quite similar in style to his earlier work The Keys of Marinus - a series of short encounters linked by an overall story arc. But while the linking device there was a quest, to find the keys of Marinus, here it is a chase - hence, in both instances, the name.
Because the TARDIS crew spend most of their time trying to evade the Dalek pursuers, and because the Daleks have orders to kill the time-travellers, there is not much opportunity for the two sides to interact, so they instead spend most of their time interacting with the various sub-plots - first the Doctor and Companions meet Dracula (to take one example) and react to him, and then later the Daleks arrive and their response to the same situation contrasts them against our heroes.
This makes The Chase a unique Dalek story, with many good moments. Plus it introduces two of the best Companions that Doctor Who has ever had, and says goodbye to two of the original characters, without whom the series would not have been what it was. The Chase is also, therefore, part of an important transitional period in Doctor Who's history, one that left only the Doctor as the ongoing, continuous character.
It successfully passes on the Companion baton from Ian and Barbara to Steven and Hi Fi. Looked at that way, the presence of the Daleks in The Chase is almost the least important thing about it.