When I opened up my computer what I saw was that one gif of Ian Chesterton rocking out to Ticket to Ride from the very beginning of "the Chase." You guys have no idea how much better that made my day. I need to keep that window open for ever.
So, in that scene, they've just picked up the Time TV, and Ian and Barbara have been having fun looking at history without being arrested by the locals, and Vicki, full of teenage squee, runs over and pops in her selection. And it's this absolutely contemporary Beatles concert. And Ian and Barbara recognize it, but recognize it as something they haven't seen, and in a weird mixture of awe and horror ask her what year it is. And she exclaims happily "1964!"
Now, they left earth (read: were kidnapped) in 1963 and have been traveling for over a year. This is their immediate future - or, alternatively, their present that they've been missing. It's a totally unexpected little slice of home, a very "now" moment. And they both kind of stop - you can see them being forcibly hit by this. And then they're like "screw it, we're gonna have fun" and just start rocking out. Like, all four of them, crazy cantankerous grandpa included. And here's Ian, dancing like an idiot and totally not caring because they're all just enjoying this one perfect moment. It's the Beatles, it's 1964, and they're British - this isn't some distant thing, this is very much theirs.
And then it's over and the Doctor's like "yay, my favorite Beatles!" and Ian and Barbara ask Vicki (teenager from THE FUTURE) what she thought of it - introduced to their culture after they've been having to deal with hers for a while. And she's all "They're great, but I never realized the Beatles played classical music!" And you get our heroes once more completely dumbfounded. And there's this awesome juxtaposition of past and future and immediate now, simultaneous connection and alienation with the other cast members, and of course the fact that, even though they're from different times and different planets, they're all just hanging out and enjoying this Beatles concert.
And...this is Ian and Barbara's final episode. Six half-hour segments later they're back home in 1965 or so, a home that they really never thought they would see again. And that scene in the beginning just highlighted the home-as-TARDIS vs. home-as-earth... stuff and...it might be the most amicable companion departure in the entire show and it's just beautiful.
So...yeah, that one scene in some sense is everything I love about Doctor Who. Including, but not limited to, Ian Chesterton in a stripey shirt dancing like an idiot without a care in the world.