After some more experimentation, I can report some additional information on navigating inside stars. I made a sketch to illustrate what I learnt.
Warping to a star will bring you to a point, north of the star, on the "distance 0" sphere. Everything inside this sphere reports as zero effective distance to the star. You cannot place effective bookmarks in there, you'll be brought back to the zero sphere. But you can travel with your ship inwards, up till the point you reach the "bounce zone". For New Eden's star I think it lies about 800 km inward. Once you reach the bounce radius, your ship is flung away from the star.
Though you cannot warp to a bookmark inside the star, you can warp to deployable structures inside it, for example you can drop a mobile depot, and then warp back to it from any location. From there, you would have a shorter travel time to the bounce zone, and hence a shorter time before you get ejected. If you warp to a mobile depot that is inside the bounce zone, you get insta-warped radially outwards.
So how to get a mobile depot inside the bounce zone? You have to build a "ladder". You start with placing a mobile depot outside the bounce zone. Then you travel to the opposite side of the star, and warp to within 100 km from the mobile depot. This brings to deeper into the star, and you can build the next rung.
Oh, apparently there are such bounce objects also in Jita's sun!
