From the pictures I have seen of airline simulators, it looks like the range of motion is limited to somewhat level flight. Are there any flight simulators, possibly military trainers, that can do a complete barrel roll?
Pulseworks makes some full motion sims that are capable of inverted flight and infinite barrel rolls. These sims fall into the entertainment category more than the training category, but they are full motion, in cockpit sims. I have also been in a similar type of sim at a Harris/Boeing sponsored party in Seattle made by a company whose name I cannot remember that did the same thing. That one in fact was capable not only of rolls but also loops and could be pretty violent (I wanted to see if it would spin... it did).
Here's a video of one of the Pulseworks models. They have some at the Udvar-Hazy center at IAD (free admission, and a shuttle from the terminal). They cost a bit of money to get into. They are also often setup in pairs so you can go into a dogfighting scenario with the box next to you. Bring a friend.
It is possible to design one which does but impractical.
The PulseWorks simulator, while entertaining, is not realistic and cannot accurately simulate inverted flight. The simulators motion is not made so the sim cabin matches the actual attitude of the real aircraft during maneuvers; rather to induce sensory illusions in the trainee crew which match what one would expect in flight.
Full Motion Simulators (FMS) are used, primarily for transition and currency training on large aircraft which operate in the air transport category and where the training is simply impractical for both cost and safety reasons to do it in the real thing. They are designed to create motion which can mimic sensory illusions during flight in the +1-1.3 G range, typical for all commercial air carrier or cargo operations. As such building a simulator with full inverted capability would simply be unwarranted for the requirements of a typical FMS. It also provides a pretty inaccurate simulation of the kinesthetics of inverted flight, as most inverted condition end up being in the positive G range. It's also not practical for aerobatic or military flight training as at most it could only simulate -1G and -1-2G for a brief moment, and the whole simulator module would have to be rolled inverted for the effect even when simulating negative G maneuvers while the aircraft is erect.
Like it has been said here it is very difficult to simulate g-forces outside of the common civil flight envelope.
Like Zeus said, even with this 2DOF 360° you will feel "wrong" g's very often, e.g., banking with 90° feels like a 1 g rudder slide. And a loop feels more like flying inverted (-1 g). But in real life you have +3 g's.
I think this simulator is very useful to practice aerobatics!
You feel the same emotions, fear, and excitement, like in real life, e.g., my first real life loop scared me in the same way the simulator did. Same thing with barrel rolls, lazy eights, cubans, hammer heads, inverted, etc.
Also it is very exhausting and your stomach will have some fun for sure :-D Flying with a VR headset teaches the right reactions very, very well. Actually I make, or do not make, the same mistakes both in the simulator and in real life.
A friend (a Lufthansa A320 pilot) was also surprised how good this thing is in teaching, but to reach this level the simulator requires many little (invisible) additions: sound shakers, g-effects (on set cueing), VR headset, g-belt tensioner, force feedback, and many things more.
In comparison to standard simulators, this concept is far advanced for aerobatics or military training, IMHO.
I believe there was one at RAFC Cranwell that could go inverted. Source: my sister's, at the time, fiancée who was training there. I may be remembering wrong - it was a while ago.
Also, while not a flight sim in itself, many years ago there was an arcade cabinet for Sega's G-LOC called the R360 that had full 360 degree motion about all axis. I would be gob-smacked if that same tech had not been used in more professional environments.