In the original paradox both the door close for a brief period of time and then open simultaneously in rest frame; whereas according to the ladder's frame both doors do not close simultaneously.
But what if in barn's frame we shut the door simultaneously and do not open them at all while the ladder is inside?
Will the ladder be inside?
There is no difference if doors are kept open or closed, from the point of view of the contraction of the ladder's length in the barn reference. Maybe the question is more about what would happen to the ladder if the doors are both closed as soon it is inside the barn.
The head part of the ladder will smash against the closed door. In a purely ideal case of perfectly elastic materials, the door and ladder will start to be deformed. Deformation of the ladder will not be confined to the front part's neighborhood because a compression wave will start to travel towards the tail. However, the speed of deformation is much smaller than the original speed of the ladder and the part not reached by this wave will continue with the original speed. It would be possible to analyze such a theoretical scenario but, even though the relativistic ladder is already a non-realistic situation. the perfectly elastic material would add some more implausibility. Already for a high but non-relativistic speed, there is a limit to elastic deformations beyond which defects will trigger cracks that will destroy the materials (both of the ladder and the door).
Considering the huge kinetic energy of a relativistic ladder, it easy to predict that ladder, doors, barn, and all-around will be vaporized in a huge explosion. I leave as an exercise to evaluate the energy released by a 20 kg ladder hitting the door at a speed corresponding to $\gamma=2$.
Your question is self-contradictory. You ask if the ladder will be inside after keeping both the doors shut while the ladder is inside, so you are already assuming the answer.
The pole is too long to be kept inside the barn. Its leading end will collide with the closed exit door. In the pole's frame the collision will happen before the entry door is closed.
Marco Ocram's answer is correct and shows there is no paradox in the pole's rest frame. If this is Lorentz transformed to the rest system of the hangar, with the exit door not opening, the surprising result is that the pole starts to lengthen (its back end slowing down) before it hits the closed door, so that it has its rest frame length when it stops.
We imagine the ladder moving from left to right.
The problem here is that objects are not infinitely rigid. We model the barn doors and the ladder as springs. However, we'll model the barn doors are much stiffer springs than the ladder so we can treat them, again, as infinitely rigid.
So we now think of the ladder as a spring. In both the ladder frame and the barn frame the spring is at its natural extension length even though this length is different in the two frames.
In the barns frame the ladder-spring will enter the barn, and there is a span of time when the ladder is wholly contained within the barn. Suppose when this span of time starts both barn doors are closed and remain closed.
After a short period of time the ladder will hit the right wall of the barn and the right end of the ladder spring will stop instantaneously. The left end of the ladder will still keep moving to the right because no forces have yet acted on it.
There will then be a period of time where the size of the ladder is shrinking even more as the right end is stuck against the right barn door, but the left end is still travelling right. However, now the spring is compressing below its equilibrium length so there starts to appear a leftward force slowing down the left end of the ladder.
This is where the picture gets quantitatively fuzzy for me, I don't really know how to describe relativistic springs. I also have a hard time describing accelerations, even in special relativity, since I'm not an expert.
The final state of the system will be a ladder which is not moving and whose right end is stuck against the right wall of the barn and whose left end is stuck against the left wall of the barn. The spring is now compressed because, with the spring at rest, its new equilibrium length is longer than the barn.
How exactly it gets to this state I'm not clear on. In the ladder's frame of reference (which is non-inertial...) it begins at its equilibrium length, it then has a barn wall fly into it at high velocity which must compress it, then it begins to re-expand, but before it can get back to its equilibrium length it gets squished in by the left barn wall from behind.
But this is the story and resolution to the paradox. In short: The ladder-spring crashes into the right wall and a sound wave propagates through the ladder-spring causing it to slow down and compress, with the end result being a compressed spring trapped between the two barn walls.
Team barn will see the ladder stretched in length because the light from the front has less distance to go, so it will be seen moving before the back, which will be seen where it was a bit further back in time. Team ladder will see the barn coming towards them (like scenery passing a motorist). The barn will appear longer to them for the same reason. Team barn will see it won't fit, but team ladder think it will and it will stick out one, or both of the doors. The Lorentz Contraction effect (calculated with Pythagoras's Theorem) occurs when viewed at 90 deg. to direction of motion. They wouldn't get to close both doors, because the ladder would be in the way. It only appears shorter from behind or the side, but is actually the same length.
Example of Real Situation: You are viewing a galaxy about the same size as ours (100,000 light years across) facing almost edge on. The light from the far edge has been travelling for 100,000 years longer than the light from the near edge, so the light emitted at the same instant won't get here for another 100,000 years. If it is travelling at 1 billion km a year, you will be seeing the far edge where it was 100,000 years earlier than the near edge. This will make the near edge appear 100,000 billion km (10.56 light years) further advanced in the direction of motion. Not much in galactic terms (about double the distance between 2 adjacent stars), but it will appear imperceptibly distorted. If it was travelling at a trillion km a year towards us, by the time the light from the far side reaches here, our descendants will have been seeing the near side move 10.56% of the diameter closer, so it will appear lengthened by 10.56% in our direction and the distortion may be noticeable. There are some Newtonian laws that even light cannot break, and taking longer to get from further away, including the far end of an object is one of them.
