Charged antimatter particles are stored using electric and magnetic fields in near vacuum conditions. (Near-vacuum conditions can be created on Earth)
Anti-hydrogen is stored by exploiting its magnetic properties. (While neutral, it still has spin magnetic moment. The storage is done using strong superconducting magnets.)
Antiparticles are easier to store when they are low energy. One of the biggest problems with storing antimatter is the high energy and velocity they are generally created at. It helps to slow them down. (for instance, Anti-protons are made to pass through a dense electron gas, which slows them down.)
The maximum amount of time anti-hydrogen has been stored on Earth is 16 minutes. (June 2011, ALPHA at CERN.)
The creation of anti-hydrogen is easy because it only involves the combination of an anti-proton and positron. It is much more technical and difficult to create higher/heavier atoms, even anti-helium.
See this link for more information: http://home.web.cern.ch/about/engineering/storing-antimatter
Paper on observation of antihelium-4 nuclei. (The difficulties become obvious): http://arxiv.org/abs/1103.3312