Not saying it's impossible as such, but am convinced that we won't see transporters in our lifetimes.
As someome else said earlier, I won't discount it flat-out; no-one knows what the future holds, but I don't beleive that the tech to disassemble and reassemble something as complex as a human body, on a molecular level, will exist in my lifetime, and probably not my children's either.
I think transporters are impossible. Converting you to energy just means that the machine is killing you. It's killing you instantaneously and painlessly, but it's still killing you. It then makes a copy of yourself at a new location. Is it really you? Does it contain your soul? I think it's an evil twin.
Wormholes, or Iconian Gates, would be a better alternative to instantaneous travel. These don't tamper with or kill your body when you use them.
Actually the only thing I can think of as impossible are the lack of crew safety equipment on starships. No bars to hold on to, no seat belts, not cushioned walls. Space exploration for real the way they build those interiors can injure someone seriously in the event of battle, speed, collision, or just plain turbulence as seen in so many episodes and series.
The rest is doable, given enough time and energy and knowledge devoted to developing those systems.
On an unrelated note, I'm searching for longterm active players like myself to build a longterm fleet with me. If you haven't heard it, it's all the rage in zone chat. Check it out in the Starbase One section, title of the thread is: STO Talent Search. Hope this reaches the right people.