CURRENT CREW APPROACH:
The solution is easy if you think inside the Box <-- is this still a pun?? Just copycat and the first person to convert the anime shares it with everyone else.
CURRENT CREW APPROACH #2:
You mentioned that the getting of converted of movies viewable on an iPad was your main problem. If anything, if someone bought a portable WIRELESS hard drive and used that to archive a copy of every movie and anime converted or downloaded by everyone, then that would make sharing things amongst the crew much easier and people would start to join your little video group so to speak. Of course, if you are forbidden EM devices, then a Wi-Fi external drive is no good. Then it wouldn't be so tedious going from one person to another to find out what is available because everyone will be copying their stuff into the wireless storage drive.
SUGGESTION #1:
Since this is Bakabt, all good idiots like us are thinking outside the box. So, since a laptop is clearly superior in terms of what it can do compared to a proprietary iPad system, what is stopping you from sticking the laptop to the low ceiling above your head? If opened up, the total exposed area would have enough velcro real estate to support the weight of the darn thing without it falling on your head when the ship plows thru those 30 footers. As insurance against falling, if you are able to get two _thin_ fabric straps wrapped around the overhead rack, they can be positioned to provide insurance or even add support for the laptop. These must be a thin strap for reason mentioned later. Now imagine the laptop already velcroed to the ceiling. One thin strap is across the main body of the laptop just under the hinge and above the keyboard. This supports the top end of the heavy section of the laptop. The other strap lies across the bottom area of the laptop probably over the area with the trackpad on it. The reason why they must be straps is that if the laptop ever falls down around one of the straps, the weight will presumably cause the clamshell to close somewhat. If you had used a wire or rope, the thickness would cause the lid to crack because there is not much clearance when the clamshell begins to close up. To prevent the straps from becoming separated to the point where they don't support or give insurance, you will have two wooded furring strips or think dowling. One piece of wood on each side of the laptop and each end of the piece of wood is fastened to one of the two straps. That way, the straps cannot move either closer or farther apart. Also, if the laptop decides to slide sideways, the wooden piece will stop it from moving to the side. The wood can be fastened to the straps by drilling a hole through it and the strap and then threading string through the holes to tie it to the strap without causing the strap to bunch up too much. That's why you drill or poke a hole in the strap and tie the string around that small cross-section of strap instead of the whole strap.
Another advantage of straps is that they cause no damage to government property and thus it meets rules requirement 510.21 but I am not sure what the Chief in your area would say if he saw the few inches of strap. Perhaps you can spray paint it white; spray paint is thin so it should not crack when flexed some. Brushed on types of paint are too thick so they may crack off.
SUGGESTION #2:
If you have a power outlet at the berth, and if the straps are allowed by the Chief, then consider three straps. These straps would be fastened to a thin rectangular piece of plywood about one foot by two-and-a-half feet. The wider part of the plywood brings that edge of the wooden piece closer to the sides of the berth so that there will be less downward droop due to the weight pulling downwards on the strap. Make the plywood as close to the edges of the berth as reasonable, but the other width doesn't need to be much more than a foot. Now, buy a nice LED backlit LCD monitor. The LED backlit ones are thinner than the older style mercury lamp ones and will maintain a better whiteness level as the monitor gets older. You will also buy one of the thin low profile VESA wall-mounts for the LCD monitor. VESA refers to the standard that defines where the mounting holes are on the back of the monitor. You will mount the monitor to the plywood rectangle. You may have to buy a "right angle" power plug for the monitor unless it is one of those that doesn't have the plug come out of the back of the monitor perpendicular to the screen. My own 1920 x 1080 LCD monitor has the plug go downwards so it is conveniet to mount close to a wall. If you do this, you will probably want to do it right because if you are the first person on the ship, the XO might take a tour to your berth just to see what this sailor is up to since the gossip vine will prolly exaggerate your monitor setup. Only you know how big the swells are and that will dictate the size of monitor you dare to have. Of course, if this is deemed a hazard, you will not be able to do it. However, this is BAKAbt so we want to have the best and most extravagant practical and do-able ideas outside of the Box.
SUGGESTION #3:
If you are allowed to drill holes in the rack to mount something, much of the above rigmarole is unnecessary. But I suppose the officers do not allow ANYTHING to be modified. I don't know what all the naval regulations are with regards to berthing.
CONS:
Laptops want external power so if you don't have an outlet in reach of your bunk, that sucks. Laptop batteries are too expensive to churn through. Velcro should not cover hot spots on the bottom of the laptop. If the straps idea works well enough for you, you might even not velcro the bottom of the laptop which would cause it to have an airgap thus helping cool the bottom of the laptop. The velcro on the lid would stop the laptop from sliding backwards, or you can add a stop to the bottom strap. Sometimes, an idea that would otherwise be allowable for a single person to have might be disallowed by the officers simply because everyone else will do the same thing if they allow that one person to do it.
Okay, I've typed enough.