As What-If #58 references, it's quite popular to try and come up with ways of getting into space differently than the way we do it now. Tons of the "why don't we send a rocket up to space and THEN go sideways", which of course misses the point of how carefully launch trajectories are designed to be as efficient as possible. Threads like this one, for example.
Space guns are a great idea -- it's way cheaper if most of the launch energy comes from the ground rather than being carried with you, and you could use a high-efficiency high-reusability electromagnetic launcher -- but they are the worst in terms of drag. Having maximum velocity at the start means you lose most of it to drag.
Of course, there's the combination -- why not lift a space gun into space and then fire a payload up to orbital velocity? But the cost of lifting a heavy space gun and all its fuel is prohibitive and wasteful. If only there was a way to have the space gun already up there....
First, launch 400 robust low-resistance metal rings into orbit. Each ring will need to be about 8 meters across. Bolt them together with a robust, lightweight frame, around 10 meters apart to form a tube around 4 km long. Yes, that is a very large object to have in orbit. Yes, this will be a very high initial cost. Not much we can do about that. Of course, the materials and design elements themselves will be nowhere close to the cost of the Large Hadron Collider (which is a loop seven times longer than the tube we're talking about); the highest cost is going to be getting everything into orbit and putting it together once it's up there.
Add solar panels, high-efficiency solar ion engines, and a central control center with a habitable module.
Wire this tube such that a high burst of current can be sent through each of the rings in sequence, turning the whole thing into a gigantic solenoid with a uniform magnetic field through the center.
Now that the orbital accelerator is all set up, all we need to do is set up a payload!
Design and build a fully-reusable mid-range rocket capable of suborbital space flight and re-entry with a peak speed of around 4 km/s. This level of performance was achieved in the 60s with the Jupiter class of medium-range ballistic missiles; modern improvements should make it quite easy and inexpensive. Mass fraction would be significantly lower than orbital launch; consider a mass fraction of 5:1 using SpaceX's Merlin-class engine vs a mass fraction of 28:1 for orbital launch with the same engine. Using a peak-efficiency liquid rocket could reduce mass fraction as low as 3:1. It may be that this is low enough for air-launch systems to be cheaper and more efficient, but that's another question entirely.
Incorporate a lightweight, robust electromagnet design into the rocket's outer skin. Also, give it some reusable retro-rockets for ultra-fine flight control.
The launch trajectory will be designed to intersect the orbital path of the space-station-tube. High-speed computers will determine the exact trajectory of both spacecraft and line them up such that the tube catches up to the rocket and passes around it from back to front. At this point, both the tube solenoid and the rocket's electromagnet are activated, exerting tremendous force on the rocket and thus transferring momentum from the tube to the rocket. Acceleration on the rocket will be on the order of 200 gees, boosting it from 4 km/s well up to orbital velocity.
Because the mass of the orbiting coilgun is so much greater, it will lose only a small amount of speed in the exchange and can make up the difference with a very short station-keeping burn of its high-specific-impulse ion engines.
The returning spacecraft can execute the exact same maneuver in reverse when it wants to de-orbit, dropping from orbital velocity down to 4 km/s. This should be enough to greatly reduce heating on re-entry and enable a parachute-assisted descent with pinpoint vertical landing.
In this case, we'd need the coilgun to exert a force of 50 MN. Note that this would produce an acceleration on the 2800-tonne spacecraft (see below) of a mere 0.14 gees, reducing the station's speed by only 106 m/s. This is still around 4 tonnes of propellant, but that beats the heck of out the ~250 tonnes of extra propellant that our rocket would require to achieve orbital velocity on its own.
Cost. If we assume that the conductive rings will each be about 10 cm thick, then they'll each weigh on the order of 3.5 tonnes. SpaceX claims a target launch cost of $500 per pound, which means that if the rings compose half the mass of the spacecraft (the rest being frame, engines, modules, and so forth), we're looking at a total mass of 2800 tonnes and a launch cost of $3.1 billion. On the plus side, the cost of the materials is going to be negligible in comparison, and this is less than half the cost of the Large Hadron Collider and nowhere near the pricetag of the ISS.
Power. Achieving rapid enough energy discharge into both the rocket's electromagnet and the orbital coil will be tricky. For this purpose, a flywheel may actually be the best way to store and discharge electrical energy from the solar panels on the space station. Powering the electromagnet in the rocket is trickier, though a flywheel could certainly do the trick here as well.