I have read articles and seen videos explaining why an SSTO (Single Stage To Orbit) rocket* is not possible. But I was wondering... What would be required to achieve this? Answers can be literally anything.
*: I am referring to rockets capable of taking supplies and humans to other planets.
I am referring to rockets capable of taking supplies and humans to other planets.
For an interplanetary single-stage rocket with tens to hundreds of tons of payload capability, no existing propulsion system can do the job in a practical way. Chemical rockets lack the fuel efficiency; electric rockets don't have the thrust required to leave Earth's surface. Even the existing designs for nuclear-thermal engines (both the US and USSR developed working prototypes in the 1960s) don't have either the efficiency or the thrust.
To do it, we would need a vastly more powerful and efficient rocket engine. The "nuclear lightbulb" engine is the next step up in capability, but it has a lot of unsolved engineering challenges and likely wouldn't have the thrust-to-weight ratio needed to get itself, its fuel, and a useful payload off of Earth. Some sort of fusion rocket could probably do it, but we don't even know how to build a working sustained fusion reactor on the ground yet.
Rockets are basically devices which exploit Newton's Third Law, for every force there is an equal and opposite force. By throwing mass out the back as fast as possible this imparts an equal force that lifts the rocket, engines, payload, and all its own fuel.
Single-Stage-to-Orbit can be done, but it's horribly inefficient. This is because of the Tyranny Of The Rocket Equation which basically says in order to get more mass into space you need more fuel which means more mass which means more fuel...
You can calculate the fraction of a rocket which will be fuel with the simplified equation $\text{fuel fraction} = 1 - e^{-\delta v/v^e}$ where $\delta v$ is the desired change in velocity and $v^e$ is your exhaust velocity. The faster you throw mass (ie. fuel) out the back of your rocket, the more force it imparts, and the more efficient it is.
We can work out the $\delta v$ from the surface of the Earth to low Mars orbit using this awesome Delta-V Map. It's about 15 km/s.
The most efficient conventional chemical engines burning hydrogen-oxygen have an exhaust velocity of about 4.5 km/s. Plug that in and we get 96.4% of the mass must be fuel or about 28:1. An empty Falcon 9 weighs about 28,000 kg, and an empty Dragon capsule is about 4,200 kg. To just get this 32,000 kg vehicle to Mars, no people, no supplies, would take about 900,000 kg of fuel or about 2 to 3 times what a Falcon 9 can hold. A Falcon Heavy can hold that much fuel, but now you have three times as many rockets or 88,000 kg. More mass means more fuel. 2.4 million kg of fuel, about twice as much as the Falcon Heavy can carry.
There's three options.
SpaceX is packing in more fuel by keeping their tanks as cold, and thus dense, as possible right up to the moment of launch. This raised a safety problem, people are normally loaded onto rockets after fueling just in case something goes wrong during the fueling process. SpaceX wants to "load and go" meaning they load the people and cargo, then fuel and launch asap before their fuel heats up, expands, and leaks out. SpaceX got NASA approval, but fuel tanks are about as dense and large as they're going to get. This won't get you to space in a single stage.
Let's say instead of a chemical rocket you used a nuclear-thermal rocket with an exhaust velocity of 9 km/s. Suddenly to get our 15 km/s we only need 81% fuel or about 5:1 ratio. Now getting our 32,000 kg empty Falcon 9 + Dragon to Mars is only 160,000 kg of fuel. Another 100,000 kg of fuel gets you 20,000 kg of payload.
Of course there's the pesky problem of radioactive exhaust (solvable) and spewing radioactive material if the rocket explodes which sometimes happens.
Even higher $\delta v$ can be had from Ion Thrusters of 20 to 50 km/s. Take this to the extreme and you have the photon thruster whose exhaust velocity is the speed of light. Trouble is these produce such anemic thrust they can't lift any appreciable payload against Earth's gravity and push through its atmosphere.
So, for the time being, we're stuck with 4.5 km/s to get to orbit. Once in orbit we can use more efficient, or more dangerous, engines. But that's staging which brings us to why we have multi-stage rockets.
The rocket equation tells us that every kilogram we shed we gain back many kilograms of payload. If we shed it early in the launch, that's less mass we need to keep pushing along.
It turns out empty fuel tanks weigh a lot. So do the many engines needed to push through Earth's atmosphere. Once you're above the atmosphere, above most of the drag, you need a lot less thrust to get to orbital velocity. So we dump them as soon as possible. This is the first stage.
Going back to the rocket equation, say the first stage only needs to produce 5 km/s, then it's discarded. At 4.5 km/s this give us a fraction of 67% or 3:1, very good! Then the empty first stage is jettisoned, about 8% of the launch mass. This leaves us with 25% of the mass. This second stage must provide the remaining 10 km/s. This means the remaining 25% of the original mass must be 89% fuel or 22% of the original mass. 67% + 22% = 89% or about 10:1. This is a vast improvement on single stage which was 28:1!
In reality this would probably be a three stage rocket to achieve even more efficiency. And this is why we stage rockets.
The Holy Grail of space propulsion is to escape the Rocket Equation entirely and not have to carry your fuel with you at all. Probes of the inner solar system can use solar panels to power their ion engines. They still need to carry reaction mass, but they don't have to carry fuel to propel it. Instead they use sunlight to generate electricity to propel their fuel at extreme velocities and make the most efficient use of their reaction mass.
This is currently in use. For example, Dawn has about 425 kg of xenon (chosen because it is non-reactive and very dense) for reaction mass, but it uses solar panels to propel it. Those solar panels provide about 1kW at its target Ceres. This gives it an incredible $\delta v$ of about 10 km/s, but it takes about 4 days to go from 0 to 100 kph. No good to lift off from the Earth, but great in the vacuum of space.
Similarly a laser thermal rocket uses a catapult, conventional rocket, or aircraft to loft it high into the atmosphere. Then ground and space based lasers heat the rocket causing its fuel to be exhausted at higher velocities than are achievable by conventional rockets.
This is entirely speculative.
I think the SABRE hybrid air-breathing engine/rocket engine and spaceplanes are worth mentioning here.
In order to have a vehicle capable of achieving orbit, it should be obvious that you need an engine capable of operating in a vacuum. One of the issues with vacuum engines is that their efficiency is way lower when in a dense atmosphere, and thus (as Schwern points out in another answer) they have to eat through tons and tons of fuel to deliver the required Delta-V (and that's if they can deliver the needed thrust to counteract the gravity drag at launch).
The classic approach to the problem is to use different rocket engines at different altitudes (read "different atmospherical pressures"). Take the space shuttle program: solid rocket boosters lower stage offers high thrust (12000kN) and low efficiency ($I_{SP}$ of about 250s); its main engines deliver low thrust (1800kN) and higher efficiency ($I_{SP}$ of about 450s in a vacuum).
Spaceplanes use a completely different approach: the ""lower stage"" is a plane with air-breathing engines, pretty much like modern-day airliners. A plane doesn't need high thrust to defeat gravity drag, because it can rely on aerodynamics to provide lift. And the fuel efficiency of turbofans (the kind of air-breathing engines that airliners use) is crazy when compared to rockets: ($I_{SP}$ of about 3000s). They "cheat" the "you must carry your own fuel" rule by using the oxygen in the atmosphere as part of the reaction mass (edit: and by using air as propellant; see comments).
The general idea is to use air-breathing engines to gain some altitude and speed and then, when the atmosphere is thin enough for the turbofans to not be of any more use, engage the (vacuum) rocket engines to achieve orbit.
You might remember Virgin's SpaceShipTwo: the plane part flies up to 15 kilometers, then the rocket part detaches, engages its engines, and flies up to 110km, off the atmosphere.
But you still need two different engines. That's when the SABRE hybrid air-breathing engine/rocket engine comes into play. Instead of a turbofan and a rocket engine, you have one engine that can behave as either. Instead of turning off the turbofans and firing the rocket engines, a SABRE engine would just switch modes at the right altitude/speed.
Unfortunately such a hybrid engine is a very complex engineering task. As of today (2018) it's just a concept without a working prototype. Quoting from wikipedia:
In 2012, [the manufacturer] expected test flights by 2020, and operational flights by 2030.
So I think it might be possible to see low-payload or passenger-only SSTO spaceplanes (but not SSTO rockets) in two to four decades.
If you're only interested in a safe reusable SSTO to low earth orbit with a small payload ratio, you're in luck (kind of). The acetalyne-ozone fuel mixture would give both the ISP and the thrust to pull it off with otherwise conventional technology. You have two problems to solve:
1) Figure out how to stabilize 100% liquid ozone.
2) Figure out how to build an engine that can tolerate the heat of combustion. Historically, this was done with liquid oxygen cooling. You cannot use liquid ozone cooling.
