Would it be possible to create a hadron collider in space? Would it be theoretically possible to create a hadron collider in space by orbiting particles around a planet?
 A: Not by orbiting particles around a planet. The LHC needs a lot pf precision to work. The proton beams need to travel along the exact path we tell them to, and they must collide in a precise area. In space, due to the gravitational field being nonuniform (nearby massive bodies will cause perturbations which will change with time and be impossible to circumvent), we really can't do much about that. Aside from that, there are a lot of high energy particles traveling through space already (solar wind/etc).
That aside, we have no way of accelerating the particles. Particle colliders send the beams on a circular path. They are continuously accelerated, and after some number of laps, they are made to collide with a reverse beam. Orbiting particles can't be easily accelerated, without adding a superconducting ring. And, if we have a superconducting magnet ring, then we don't need the planet in the first place--we can keep the system anywhere.
In short a large particle collider can be built in space if we can amass the necessary resources -- but utilizing a planet's gravity for anything other than keeping the entire collider in orbit is a fools' errand -- so using the orbits is out of the question.
A: The reason we need a large radius for the colliders, LHC has a perimeter of 27km, is so that we can reach a very high energy in a controlled manner, with the type of magnets and cryogenics our technology can devise.

In addition, as the energy goes higher the circulating charged beams start to emit synchrotron radiation  to such a large extent that the control of the beam is lost. Note that in the tunnel of the LHC the LEP experiments first ran, which collided electrons on positrons not much over 100 GeV. Radiation is the reason that now the next lepton collider, ILC, is planned as a linear collider.Higher energies for electrons in a circular accelerator are self defeating as the energy fed is lost as synchrotron radiation instead of accelerating the leptons. The LHC ring can accelerate protons to TeV energies because the proton mass is so much larger than the electron one. There is a limit to the energies that can be usefully achieved again because the losses from radiation will not allow high energies to grow effectively.
A large ring in space around the earth would would have the advantage for allowing  very high energies to be achieved, with small radiative losses as they scale inversely with radius (this is for electrons) :

Why around the earth?
1) Satellite technology is very advanced and the satellites accurately positioned. One would need to set the quadrupoles and dipoles in satellites  to keep the beams in a circular orbit and have them collide at a certain spot.
2) cryogenics would be unnecessary  as well as maintaining a vacuum since these are available for free
So, imo,  it can be done, but the cost would be prohibitive and then to set up experiments , which would need their own satellite, around the collision point would not be an easy thing considering the number of people who babysit experiments now ( about 3000)
Maybe in the far future when nanotechnology and robotics come into their own one could have such a successful experiment.
edit in answer to a recent comment:
Chris:The LHC's vacuum is much better than that in the neighborhood of earth, though. I wonder how bad of problems that would cause?
Anna:Here is a  discussion of the LHC vacuum, the beam pipes have " a vacuum almost as rarefied as that found on the surface of the Moon. " so maybe satellites for an accelerator  around the moon ? 
A: The energy of a proton orbiting a planet is rather small to speak of a "collider". 
Let us see: if $v = 10\;km/s$, then $$\frac{M_p v^2}{2}=\frac{1}{2}M_pc^2(\frac{v}{c})^2=\frac{0.938\cdot10^9\;eV}{2} \cdot (\frac{10^4}{3\cdot10^8})^2 \approx 0.52\; eV$$
Besides, magnetic field will affect charged hadrons and prevent them from "orbiting" a planet.
A: This idea, accelerator encircling Earth, was the subject of 1954 lecture by Nobel laureate Enrico Fermi. The account of this 'ultimate accelerator' could be found in the book

Orear, Jay. Enrico Fermi - The Master Scientist. 2004, abstract, pdf, (pp. 37-41).

From the book:

Next  Fermi  makes  a  preliminary  design  for  a  single  ring  proton  accelerator  of  energy $ 5  \times  10^{9}\,\text{BeV}$

Nonstandard unit $\text{BeV}$ is the same as $\text{GeV}$. For example, accelerator Bevatron, which started operating in 1954, is an acronym for "Billions of eV Synchrotron"
Image from the book, originally a slide of Fermi's talk:

Further excerpt (bold face is Fermi's notes for the lecture):

“Preliminary design...8000 km, 20,000 gauss” Such a single ring would give the desired energy, but the radius of 8,000 km or 5,000 mi would put the orbit 1,000 mi above the surface of the earth! This is shown in Fermi’s Slide 3 as our Figure 15. The entire ring magnet would be in orbit around the earth. By now the audience must have been in hysterics. They were still talking about it when they came back to Chicago. “What we can learn impossible to guess...main element surprise...some things look for but see others (this is the same as what Feynman was saying but Feynman was more poetical about it) ...Look for multiple production...antinucleons...strange  particles...puzzle  of  long  lifetimes...large  angular  momentum?...double  formation? (now called associated production)
  At present more probable...”

Fermi was overly optimistic in his extrapolation of humanity's space capability: the estimated date for such project was 1994 (40 years into the (then) future), with a cost of 170 B$. Of course, even presently, 24 years after proposed date, we haven't come close to making such an installation.
Another point to note: the design of Fermi is a fixed target setup, rather than collider. That same physics could be accessed with a considerably less energy in colliding beams setup (just a factor of about 10 over LHC).
A: Use as a rule of thumb: 1 Tesla.m integral field gives transversal momentum of 0.3 Gev/c.
Which means that bending of beam of 10 TeV into circle we need ~210 Tesla.km of magnetic field. This translate into:

*

*105 km of 2 Tesla of magnetic field (permanent magnets are ok) or

*21 km of 10 Tesla of magnetic field with active magnets, but these require a lot of electric energy, cooling etc etc hence power station (perhaps, nuclear power station) orbiting around the Earth. By the way, cooling in space is tedious task, because at Earth one can use water, air, ground to dump the energy. In space there is only vacuum with only way to cool down is to radiate. Even at the Moon cooling is easier (dump into ground).

