What closed 3D space looks and behaves like? (Relativistic Black Hole)

Question

So I wanted to ask a question that is a copy of Why can't you escape a black hole?
From the answers, the conclusion I draw is: it's impossible to escape a black hole.

any trajectory inside the event horizon is only pointing down. That's right, everywhere you look, you look down towards the center.

I realize that I can't imagine a 3D space that is curved onto itself. There has to be a vector connecting a dot with a center of the black hole, and opposite direction must lead further away from the black hole. Few questions that could help build up this intuition:

Is there a 2D example of such space?
Is the boundary of this closed space abrupt? like one nm above the event horizon there was still an "up" direction, and then it's suddenly gone?
Does the inside of black hole has finite volume? what is it's volume?
it should still be possible to see the light falling into the black hole from outside, where this light would appear to be coming from?

I have found following contradictory in the answers to the mentioned question:

no local experiment can determine if you have crossed the event horizon.
it's not possible to get any further from the black hole once you have crossed the event horizon.

Does this imply that you can't measure the distance anymore?

Jerry Schirmer · Answer

Considering that, in the Schwarzshild interior, the time and space directions are flipped, a 2D analogy that might be more instructive is to imagine a globe, where your latitude coordinate is time, and your longitude coordinate is space.
Start at the equator at $t=0$.  You're free to travel around the circle that is your spatial part of the world, but you notice that inevitably, the world is getting smaller, and everything is getting closer together, until eventually, you hit the north pole, where the circle becomes zero radius, and directions stop making sense, and everything is compressed into infinite density.

What closed 3D space looks and behaves like? (Relativistic Black Hole)

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