Current space vehicles clearly cannot. Only the Space Shuttle survives past one use, and that's only if you ignore the various parts that fall off (intentionally!) on the way up.
You could be forgiven for thinking that space is therefore an impossibly expensive place to get to. But this need not be the case. Launch to orbit requires accelerating to Mach 26, and so it uses a lot of propellant - about 10 tons per passenger. But there's no technical reason why reusable launch vehicles couldn't come to be operated routinely, just like aircraft. The only reason why this hasn't been done yet is that launch vehicle development has been left to government space agencies. And they have had neither the priority nor the will to achieve it - they don't use even 2% of their budgets (of $25 billion per year) to study the design of launch vehicles suitable for passenger service!
So it may well turn out to be private enterprise that is the solution - plenty of ideas for reusable launch vehicles exist, and with incentives like the X-Prize, there's going to be fierce competition to see who can be first.
Space Vehicles presents some of the ideas that could change the meaning of "Space" from being a remote place where government staff carry out "missions" to being a weekend destination, just a few minutes' flight away.
Here are some key documents from the archive to get you started:
The main problem of getting to orbit is to accelerate to Mach 26 - or a speed of about 8 kilometers / second. Airliners fly at less than Mach 1, and even the Concorde flies at only just over Mach 2. So you need to go a lot faster to get to orbit. To accelerate, of course, you need to use fuel - or propellants, as rocket engineers say, because liquid propellant rocket engines use fuel and oxidizer. (Cars and aircraft don't carry oxidizer - they collect oxygen from the air. But there's no air in space.)
So you need to use a lot of propellant to get to orbit. So you need a vehicle that's mostly propellant tank. When a long-distance airliner takes off, about half the mass is fuel. But to get to orbit a rocket needs to be about 90% propellants. So all the rest of the rocket and passengers weigh only 10% - so such a rocket needs a very light structure, which is also very strong so it can survive acceleration, vibration and aerodynamic stresses. Today, most satellites are launched on "multi-stage" rockets, which simplify the problem. Only a small part of the rocket actually reaches Mach 26 and goes to orbit, so you don't have to make the whole structure so light and strong. The first and second stages can be heavier.
If you want to send the whole vehicle to orbit (like your "whole" car goes to work with you, or the "whole" airliner takes to your destination) you have to make a Single-Stage-To-Orbit or " SSTO" vehicle.
In order to make a light structure containing propellant you'd like to make it as nearly spherical as possible, because a sphere has the smallest surface-area-to-volume ratio. But a sphere's not very aerodynamic, so an SSTO launch vehicle needs to be a bit pointed at one end. And you'll see that the main designs of SSTO vehicles like SASSTO, BETA, Phoenix and Kankoh-maru are pointed, blob-shaped rockets. Not thin and stream-lined like multi-stage rockets, which can afford not to be so light-weight.
Contrary to what many people who make expendable rockets will tell you, it isn't difficult to design a "single stage to orbit" ( SSTO) rocket. In fact it's very easy - it can be done with rocket engines and propellant tanks designed, manufactured and operated 20 years ago! It's important to know this, because a lot of people will try to tell you otherwise.
A Thought Experiment
If you attach 6 SSMEs (Space Shuttle Main Engines) directly to a Space Shuttle External Tank ( ET), you could launch 30 tons payload to orbit. It wouldn't be an economical way to launch - but it's certainly possible. But please note: it's only possible taking off vertically; no-one can build a horizontal take-off SSTO.
So, suppose you built the SSME/ ET SSTO vehicle; in order to bring it back to Earth through the atmosphere in good enough shape to use again without major repairs, you'll need to add some extra equipment to it. You'll need thermal protection to prevent it melting when it comes back into the atmosphere at Mach 26 and slows down to zero! (Think of the air pressure against you if you put your hand out of the window of a speeding car. And then look at how stream-lined a Mach 2 plane is - and then think of 13 times faster than that! It's a fierce air pressure at Mach 26! And to slow down, all that "kinetic energy" has to go somewhere - and most of it turns into heat) You also need some propellant to help slow down, some legs for landing, and some control systems. And if you decide that all of this will weigh 30 tons, then you don't have any payload to orbit!
So it's tricky! Typical designs end up with 1% of the take-off mass being payload, 9% being structure, and 90% being propellant. Or 1%, 10% and 89%. Or more ambitious designs claim 2%, 9% and 89% - and so on. But you can see that it doesn't take much excess mass in the design to wipe out the 1% payload.
The bottom line is that SSTO is possible - using Vertical Take-Off and Landing ( VTOL). And fully-reusable SSTO VTOL is also possible, using modern engines, materials and other equipment - but no-one's ever tried yet. Government space agencies have budgets quite big enough to do it, but they prefer to go on doing what they already do. And companies that would like to do it find it difficult to raise the finance needed. Investors don't like risks! ( VTOHL is also possible if you really want to, but it's not a good approach, because it needs a heavier structure.)
What's not yet clear is how low the operating costs of a reusable SSTO can be reduced. And you need to get them good and low - like $1 million for a few tons of payload - which gives you passenger costs of $20,000 per person or so, in order to attract lots of passengers - in order to pay back the investment.
How low the costs will get will depend majorly on the scale of operation. If you make 50 vehicles and fly them frequently, then all the maintenance procedures etc can become routine like they are on aircraft, and their cost will drop a lo-ong way. On this point, the JRS study is interesting because it considers 8 vehicles being built every year, each flying 300 times a year, so passenger numbers are growing by 100,000 passengers per year, building up to substantial scale of operations - though still small by airline standards!
So why hasn't it been done?
The main reason we don't know how far the cost can fall is because no-one's tried yet. For various reasons the space agencies of the world spend their $25 billion per year on everything else except trying to reduce the cost of passenger transportation! And they advise against such work because they say it isn't certain of success. But you don't know what you can do until you try! Luckily, more and more work is showing that passenger transportation to space is now a feasible business proposition, and more and more business people are starting to get interested. So we're going to find out at last!