Could you actually conduct the ladder paradox problem experimentally? The very famous ladder paradox says that if you take a ladder of length $l$ and a barn of length $l + dl$ and put the ladder on a cart traveling toward the barn, you can fit the ladder inside the barn if the cart travels fast enough. This is due to length contraction, which says that a person in a non-moving reference frame will measure the length of moving objects to be shortened. However, Penrose and Terrell showed that one can not observe any such shortening when viewing a moving object. Furthermore, the object does not physically change length. So now suppose you have the technology to try and conduct the ladder paradox experiment; you have the cart that can travel up to 0.99c, you have the ladder, and you have the barn, and you want to try and fit the ladder into the barn. You start up the cart and get it going to 0.99c with the ladder attached. Since you measure the ladder to be shorter, you know you should be able to close both barn doors while the ladder is inside the barn, at some point in time. But you also know that you can't ever physically see the ladder contained inside the barn. Suppose, then, that you say that what you see isn't really what's happening at that instance; maybe at some point you see the ladder sticking out of the barn, but it is really contained fully inside the barn, and maybe you can close the doors then (you are assuming that some weird optical effects are happening while trying to find a solution to the issue). Well, that doesn't solve the problem because this "solution" implies you would have to see the doors cutting through (or hitting the ends) of the ladder, which obviously makes no sense. So the only solution that is really left is that you see the ladder fully outside the barn, you calculate that at that moment it is actually fully contained inside the barn, and you close the doors at that moment. But now what if you have cameras installed at the two sets of doors, and you have the cameras take a picture as soon as the doors close? What do they see? If they see the ladder fully contained inside the barn, then the ladder has actually become shorter, which of course can't happen. But since they can't see the ladder partially sticking out of the barn (the doors are closed!), this means they must capture an empty barn. So the ladder really isn't inside the barn at that time, which suggests that while you can measure the ladder to be fully inside the barn at some time t, it cannot actually ever be fully contained inside the barn (which could of course be seen without this whole discussion by noting that length contraction does not actually happen).
The conclusion, then, seems to be that you can measure that an event happens even though it never actually happens. Is this the correct conclusion, or am I missing something? It seems like a pretty big conclusion that isn't discussed very often, which is why I'm uncertain of its validity. I'm not sure what parts of what I said would be incorrect though. I think my arguments rest on three main assumptions; length contraction can be measured, it can't be observed, and it doesn't physically happen. I'm near certain these are correct assumptions, so I'm not sure where else my arguments fail.
 A: Yes, we really can do a version of this experiment that demonstrates that length contraction is real. Of course we can't do it with a ladder, barn, and cart, but we can do it with groups of particles in a particle accelerator. For example, proton experiments in the LHC use groups of protons (known as bunches) with a distance of about 7 metres (25 nanoseconds) between each bunch in the beam. These protons travel at ultra-relativistic speeds, with a Lorentz factor of over 7000, so the length of a bunch (and of the protons themselves) in the lab frame is less than 1/7000 of its rest frame length. Obviously, such extreme contraction has to be taken into account when planning experiments.

Furthermore, the object does not physically change length. 

It's true that the length of the ladder in its rest frame is invariant, but that doesn't mean that length contraction is merely some kind of optical illusion. It's a genuine physical phenomenon that arises because the space & time axes in one rest frame aren't parallel to the space & time axes of another rest frame in relative motion to the first frame.
The Lorentz transformations guarantee that all observers will calculate the same rest length for the ladder no matter what contracted length they measure for it in their rest frame. However, the length (and orientation of the ladder) that each observer and their camera sees is different from what they measure, due to the finite time it takes for light to travel from the various objects to the camera (which is what causes Penrose-Terrell rotation). So to say what's really happening in a given frame we need to be careful to separate those light delay effects from the effects due to spacetime axes rotation.
(Note that the human eye isn't very useful in this scenario, since a ladder moving at 0.99c travels almost 300 km in a millisecond).
Due to the light delay, a picture doesn't show what's happening at all locations in the barn simultaneously. Different parts of the picture show what was happening at different times. The usual way of dealing with that (both in thought experiments and particle collider experiments) is to have a set of cameras (with built-in clocks, with the clocks all synchronised) at different locations, with each camera only photographing what's right in front of it. That is, each camera only measures what's local to it.
So we can have a set of synchronised cameras at various locations in & around the barn, and after the ladder has passed through we can use the timestamps on each image to reconstruct what happened in the barn frame. We can have another set of synchronised cameras mounted on the ladder & cart. But those two sets of cameras cannot be synchronised with each other because they are in relative motion, so their definitions of "now" are incompatible.
Here's a diagram of spacetime axes rotation from Wikipedia's article on the relativity of simultaneity. This diagram is for a 2D spacetime, i.e., one dimension of time and one dimension of space, but that's really all you need when the motion is linear.


Events A, B, and C occur in different order depending on the motion of the observer. The white line represents a plane of simultaneity being moved from the past to the future.

Your ladder is moving at 0.99c, so it has a Lorentz factor $\gamma$ in the barn frame of almost 7.09, (~ 1/0.141), so a ladder of rest length 70 metres will just fit into a barn that's 10 metres long, with both doors shut in the barn frame. That's possible because the events of the doors being closed are not simultaneous in the ladder rest frame. In fact, in the ladder's frame, it's the barn that's length contracted, down to a mere 1.41 metres.
In each frame, the front door closes after the rear of the ladder passes it, and the back door opens before the front of the ladder reaches it. That has to be the case to avoid ladder-door collisions. If there's a collision in one frame, there must be a collision in every frame (and vice versa), otherwise we'd lose physical coherency. Relativity may be counter-intuitive from the perspective of pre-relativistic physics, but it's not that crazy. ;) A collision is a localised event in spacetime. Different observers may disagree on the spacetime coordinates of events, and as the above diagram shows, they may even disagree on the temporal order of events that are separated in space, but they all agree on which events do actually happen and which ones don't happen.
So in the ladder's frame, the back door opens in plenty of time for the front of the ladder to pass through it, and the front door doesn't close until after the rear of the ladder has passed it, by which time in the ladder frame the front of the ladder is well past the open back door.
A: 
But now what if you have cameras installed at the two sets of doors, and you have the cameras take a picture as soon as the doors close? [...] they can't see the ladder partially sticking out of the barn (the doors are closed!)

The spacetime events of the two doors closing are spacelike separated. This means that at the time either door closes, the light showing the other door closing hasn't crossed the barn yet. So both cameras will in fact record the other door open and the ladder sticking part way out of it.
A camera placed in the middle of the barn and triggered at the right moment will capture a picture of the ladder fully contained in the barn with both doors closed.
A: I think this question can be quickly answered by referring to the Wikipedia page on length contraction, specifically the experimental verification and visual effects sections. Basically, the problem involves the distance from the space occupied by the ladder. According to this page (https://en.wikipedia.org/wiki/Length_contraction), there are indirect measures of length contraction. In other words, I suspect the issue is with arguing that length contraction both can be measured and cannot be observed (which appears to be a contradiction)...
A: A quick, visual answer to your question is given by the videos of wheels at the Space Time Travel site. Imagine you had a giant wheel with rest diameter $l$ rolling into the barn, instead of a ladder. You can imagine a stationary (for you) "mini-barn" in those videos, placed between the stationary wheels.
One of the lessons (re)emphasized by Einstein and relativity is that before answering a question we must make sure that the question is meaningful and not self-contradictory. Questions such as "What's the formula for the area of the square, given its radius?" or "Is the electromagnetic force blue or red?" contain contradictions or are just meaningless.
One part of the question you ask, "What would I see (standing in such-and-such position)?" or "What would a photograph taken at that instant by a camera (positioned so-and-so) show?" is meaningful and free of contradictions, so it can be certainly answered. What I personally am not sure of are other details, for example the doors. The doors will have to move, so relativistic optical effects will concern them too. And can they be accelerated and moved fast enough, compatibly with relativity, to be seen as closed with the ladder inside? A (3+1)D or maybe just (2+1)D spacetime diagram and some calculations are necessary to first check that your full thought-experiment doesn't contain contradictions. I personally don't know the answer to this at the moment.
Let me consider a simpler thought-experiment. If the barn lacked the short sides (where the doors should be) and lacked one of the long sides (of rest length $l -\Delta l$, with $\Delta l >0$) – a sort of huge bus-stop shelter – and you were outside of it, at rest with respect to it, so that you can see inside and you're symmetrically placed with respect to its short sides, then you'd see the ladder momentarily fully inside the barn, but you'd also see it rotated, with its front-moving end rotated away from you, and its rear-moving end rotated towards you. So I imagine that you might consider this as "cheating": you can see it fully inside the barn, even if its length (at rest) is longer than the barn's (at rest), but just because it's been rotated. The videos I mentioned at the beginning show this. Compare also the pictures and calculations for the case of a cube in this Physics World article; you may imagine a ladder with its ends constantly touching some sides or edges of the cube, and deduce what you'd see.
The question "Does the object physically change length?", related to a statement you make, is likely to contain contradictions. What do you mean by "physically"? and even by "to change length"? I think that if you analyse the words in this question you'll realize that it has undefined or meaningless terms within relativity.
Apologies for not being able to give a straight answer to the meaningful questions you ask.
