Now that the silliness is out of the way, I want to approach two of the technologies in Star Trek, why they are unfeasible, and why they were necessary for the over all plot of the movie. Aka, why Kirk and Spock would not have bravely defeated Nero without:
1. Warp Drive!!!!!!
So. The idea of Faster Than Light travel is nothing unique to Star Trek. Basically every sci fci universe worth it's salt has some form of FTL woven in somewhere. The way Star Trek's works is generating large amounts of energy through the reaction of deuterium with anti deuterium (matter with antimatter), using the reaction to produce highly energetic plasma, which then was used to create a warp field, which encompassed the ship, allowing the ship to enter subspace and move at "warp speed" aka very very fast.
I see several problems here. Do you? Even ignoring the whole antimatter thing (supposedly controlled by dilithium, which is non reactive with antimatter when put inside an EM field.... wat?), there is the whole underlying issue that according to Einstein, nothing moves faster than the speed of light. It just can't happen. However.... It is undeniable that warp is central to the entire concept of a "star trek", let alone the movie Star Trek.
To explain why, lets do an example. Take the distance from Earth to the Moon. Something humanity has already traveled, so its something we can somewhat wrap our heads around. Hop in your BMW (cause who cares about emissions in space?) and drive your way to the moon (just bare with me, its an example, I'm suspending the non-relevant laws of physics) and lets see how long it takes you. Well, the distance is 384.4 million meters. That is 3.84 x 10^8 meters. Let's be generous and say you are driving your BMW on a German Autoban pointed straight at the moon, and you floor it. The BMW i8 has a top speed of 120 km/h. In meters per second that is right around 70 m/s. So you make it to the moon in.... 63.5 days.
The actual Apollo 11 mission that first made it to the moon made the trip in around 3 days. For a faster spacecraft, hop in the New Horizons spacecraft, which hit speeds of about 15730 m/s, and you make the trip in just under 7 hours.
Light would make the trip in... 1.28....... seconds.
In order to do the type of exploration and travel that happens in the Star Trek universe, where distances between stars are measured in light years, or 9.461e+15 meters, we have to move faster than the speed of light.
The other technology I have an issue with is:
The way transporters work is they essentially deconstruct a person, convert them into a data packet, shoot the data packet at a destination and recreate them on the molecular level at the destination.
Besides the obvious technological and physical issues with this tech, I want to draw attention to a moral one: if we actually did this in real life, would the person reconstructed on the other end really be the same person? Would breaking someone into their constituent molecules, turning those molecules into data, and then undoing the process on the other end produce the same living being with all the thoughts, emotions, and personality of the original?
And if the answer is yes, then it raises another question: are we more than a sum of our pieces? If someone took a collection of atoms and organized them in the exact same configuration as your body, would that new collection of molecules actually be you? It is a frightening thought, one that touches on the very nature of what makes us living creatures.
Now, as far as the plot is concerned, teleportation (as that is from a non-sciency stand point what transporters are) is probably one of the most convenient, widely useful, and profitable abilities ever. Imagine, being able to be at school, realize you forgot your super important, final grade determining paper at home. No more lame excuses, just walk over to your transporter, beam home, and grab it. On the flip side, if you are driving your spaceship full of red matter in a suicide rush at the big baddy's ship, but don't want to die, just have your friends teleport you back to your home ship, safe and sound. What could possibly be not to love about transporting?