Creating portal doors

I’m trying to achieve something like this:

The problem is, I’m getting almost there but not quite. I’m trying to render a view from a camera to texture but even though I’ve tried tens of things with my camera, something always ends up broken anyway - either the field of view isn’t right or the view gets worse as you approach it and it looks worse the closer you get to it (not to mention acute angles near the door).

How can you achieve something so perfect like in the video? Is there some precise camera setting I should use?