Before I put the lengthy comments behind the cut tag, let us just all agree that it is a real shame that Brian Epstein vetoed the idea of the Beatles appearing as themselves but much older, playing at a fiftieth anniversary concert set in, I suppose, 2013. Apparently the Fab Four were on for it but their manager was not impressed, so at least we get a rare studio clip of them playing "Ticket to Ride". This also gives us a couple of good lines, as the Doctor complains when the machine is switched off that "You've squashed my favourite Beatles", and Vicki tells us that she has "been to their memorial theatre in Liverpool… but I didn't know they played classical music!" (But how does Ian know the words to "Ticket to Ride"? After all, he left Earth the day after President Kennedy was assassinated...)
Well, to my surprise there were some more tolerably good bits. The Mechanoid city was great. The Dalek/Mechanoid battles were fun. The rapidly rotating planet Aridius was well done. The Dalek's emergence from the sand dune at the end of the first episode is pretty good. To my surprise, I even quite liked the Mary Celeste bit, though my wife and mother-in-law snorted with giggles, and the final shots of the deserted ship with the last view of the name plate was quite effective. And there was a real feeling of time passing for the characters, not just the rapid rotation of Aridius but also the meals, the Doctor and companions sleeping, things we don't often see happening.
Other good bits: The location scenes on the planet Aridius (though it also appears to be the setting for the Gettysburg Address). the scene where the other three think they've lost Vicki, and her attempts to contact them from the Daleks' time machine. Peter Purves' performance as Steven Taylor, stranded rocket pilot and Ben Gunn lookalike. Indeed all three companions are on form throughout, even Barbara playing machine guns with the Doctor's Dalek-killing device. Hartnell, when he's awake, is good, but he fluffs a number of lines and was perhaps personally upset at the departure of William Russell and Jacqueline Hill, leaving him the sole survivor of the original cast. And their departure is a rather moving moment as well.
My one complaint of Vicki/Maureen O'Brien isn't really her fault, but has to do with the crapness of three of the monsters. On four occasions she is attacked - by a Mire Beast, an Aridian, and two Fungoids - and more or less has to walk into them - I think she actually has to wind the Mire Beast's tentacle around her own neck. She pulls it off well, but the only monsters that are any good in this story are the Mechanoids, and that's not saying a lot.
Oh yeah, and the Daleks. Can't count, fall off boats, can't kill Frankenstein's monster or Dracula, easily confused by Barbara's cardigan. But this is because they are being funny, which is sort of OK but you don't want it every time. The robot Doctor I didn't mind too much, but he fluffed the crucial line which was supposed to let the companions know he was the fake - the script says he addresses Vicki as "Susan", but I missed it.
The first two episodes, on Aridius, have good settings and filming - had Dune already been published before this story was written? In magazine form, surely, but maybe not yet as a novel. But the Aridians and Mire Beasts are ludicrous. The Empire State Building scene was simply pointless. The Mary Celeste, as I said earlier, I rather liked.
The Hammer House of Horrors sequence worked rather better if the Doctor's theory was right - "we were lodged for a period in an area of human thought" - rather than it being a festival sideshow in a 1996 where the Chinese rule Ghana. But maybe that, too, is but an area of human thought. (At the very beginning Ian is reading a book called "Monsters from Outer Space - Science Fiction". The ISFDB doesn't seem to have heard of this one, so presumably it has yet to be published in Our Time Line, or else is imported from the one where the Chinese rule Ghana.)
I really hated a) the jazzy intro music, b) the time vortex shots and c) the Shakespeare meets Elizabeth I scene.
Sorry this is a bit disjointed. Lots more in-depth analysis of The Chase by Alan Stevens and Fiona Moore here, by Paul Clarke here, by Cameron Mason here, and by numerous reviewers here.