The ~200km required is daunting, but imminently possible.
A mostly (the curvature of the earth will make it point up) horizontal gun will have the problem that the projectile will leave the gun at escape velocity, at around 1 atmosphere of air pressure (either after accelerating through 1 atmosphere of air pressure, which takes lots of extra energy, or leaving a vacuum, which makes exiting the gun barrel quite a bang)
Also, your barrel will have to extremely straight.
I don't think either is 'imminently' possible.
It's physically possible for us to build a 100km tall tower with current technology. (Columns formed from many highly pressurized tanks made from boron.) It's not really economically feasible, however.
At our current level of technology, I estimate this would cost at least $200 billion in capital expenses.
A neutral-buoyancy under-ocean train between two cities otherwise un-connectable by terrestrial transit would prove some of the necessary technologies. Liverpool to Belfast would probably work. Miami to Havana to Cancun would work, if not for politics.
Relevant search terms: "vactrain", "StarTram", "transatlantic tunnel"
Other non-rocket launch technologies do seem more promising. If you could build a mass driver launcher between Galapagos and Ecuador, you could likely build a launch loop in the same spot for less capital, less operating cost, and higher capacity.