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: