How come the Doctor and Clara will never be able to see each other again?

Question

The Doctor and Clara always had fun times together but their time came to a end in Face The Raven. After their last goodbyes, the Doctor goes 4.5 billion years in the confession dial to save Clara and tells her what has happend to her:

DOCTOR: Your heartbeat. Your physical processes have been time looped. Frozen between one heartbeat and the next. Even your breathing is just a habit. You don't need it.

They they argue about whose memory will be erased:

DOCTOR: It won't hurt, it'll be nothing. You'll just pass out for a moment.

CLARA: And then?

DOCTOR: When you wake, you'll be fine.

CLARA: But

DOCTOR: Clara, just listen to me.

CLARA: Just say it. Say it. Come on. Tell me.

DOCTOR: When you wake up, you will have forgotten me. You'll have forgotten we ever even met.

CLARA: And why would I want that?

DOCTOR: Because it's the only way. That stuff in your head, the image of me, they could use it to find you.

This becomes important for both of them:

DOCTOR: I'm trying to keep you safe.

CLARA: Why? Nobody's ever safe. I've never asked you for that, ever. These have been the best years of my life, and they are mine. Tomorrow is promised to no one, Doctor, but I insist upon my past. I am entitled to that. It's mine.

DOCTOR: You and me together. Look how far I went, for fear of losing you. This has to stop. One of us has to go

Then the Doctor uses it and his memory of Clara is erased. Then he is outside, and goes to a diner to talk to a waitress about Clara. Then they both go off on new adventures in their TARDISes never to see each other again, which made me think: how come the Doctor and Clara will never be able to see each other again?

Answer 1

Because it's not safe for them to be together any more.

In their discussion at the end of the universe, the Doctor and Ashildr talk about what he's planning to do to Clara and why:

ASHILDR: What if the Hybrid wasn't one person, but two?
DOCTOR: Two?
ASHILDR: A dangerous combination of a passionate and powerful Time Lord and a young woman so very similar to him. Companions who are willing to push each other to extremes.
DOCTOR: She's my friend. She's just my friend.
ASHILDR: How did you meet her?
DOCTOR: Missy.
ASHILDR: Missy. The Master. The lover of chaos, who wants you to love it, too. She's quite the matchmaker.
DOCTOR: Clara's my friend.
ASHILDR: I know. And you're willing to risk all of Time and Space because you miss her. One wonders what the pair of you will get up to next.
DOCTOR: Nothing. Nothing at all. I know I went too far. I get it. That's why I'm doing what I'm doing.
ASHILDR: And what would that be?
DOCTOR: I'm taking her back to Earth. Somewhere safe, somewhere out of the way. I'm going to wipe her memory of every last detail of me. It'll be like our friendship never happened.
[...]
CLARA: So what happens now? Hey? Me and you, what do we do now?
DOCTOR: I'm not sure you managed to reverse the polarity. I'm not even sure that you can. It'll do something to one of us. Better than flipping a coin.
CLARA: Doctor?
DOCTOR: You and me together. Look how far I went, for fear of losing you. This has to stop. One of us has to go.

_{-- Series 9 Episode 12, Hell Bent (emphasis mine)}

Summing up that long transcript quote:

• the Doctor and Clara together form the Hybrid (an incredibly dangerous creature of prophecy)
• already they've nearly destroyed the universe in order to be together
• they can't keep on travelling together for fear of what they might do next.

---

By the way, ignore this bit:

DOCTOR: When you wake up, you will have forgotten me. You'll have forgotten we ever even met.
CLARA: And why would I want that?
DOCTOR: Because it's the only way. That stuff in your head, the image of me, they could use it to find you.

The Doctor tells Clara he's going to wipe her memory so that "they" (the Time Lords?) wouldn't be able to find her, but this contradicts the above. It's just a lie he makes up before realising she knows his real motivation because she's been watching him and Ashildr on the scanner.