I found the engineering as depicted in this movie wonky. I have already written a thread about the differences between this Enterprise and earlier Enterprises. Warp power goes - there goes the ship.
The worst flaw was the reactor core. How do people service that thing? There are no catwalks. And how does that thing work? I am incapable of understanding how that thing works. I feel there was more thought put into the earlier ships than this one.
It actually does "work". It's the Target Chamber at the National Ignition Facility in California. They were filming there for a while, in keeping with JJ's approach that he likes to film in "realistic" surroundings where possible. I think it's cool that the NIF is like a modern-day parallel to a Starship Warp Core.
I just found this great YT vid that aims to explain in simple terms how it works. When you get to the end of the animation, you'll see the target chamber, which you should recognise from the film.