While usually short-lived, isolated hydrogen atoms (also called hydrogen radicals) are well studied and chemically important species. Bond formation in simple species is always an exothermic reaction (usually highly so) with zero activation energy, which means bond formation is thermodynamically very favourable and has no kinetic barrier stopping it from happening. Simply put, the isolated atoms really want to react with each other, and they'll attract each other at a distance, so if at some moment you create a container with pure monoatomic hydrogen, the hydrogen atoms will react as soon as possible to regenerate dihydrogen molecules (hydrogen gas).
The best you can do is expose the dihydrogen molecules to energetic conditions, so that a fair amount of bonds are being constantly broken at a rate comparable to which they're formed. A simple way to do this is to strongly heat the hydrogen gas; enough thermal energy will cause the bonds to break, though the dihydrogen bond is unusually strong ($\rm{436\ kJ\ mol^{-1}}$) so even at several thousand degrees kelvin there will be only a small portion of free hydrogen atoms in the gas (see Mithoron's comment). A more interesting option is to shine a strong ultraviolet light into the gas, as the dihydrogen molecules can separate into their constituent atoms upon absorption of photons with energy around $\rm{4.52\ eV}$, which corresponds to a photon wavelength of $\rm{274\ nm}$.
Any substance can have its bonds broken through the above two methods (among others), so the same applies to all other diatomic molecules. The stronger the bond between the atoms, the more uncommon is it to find the isolated atoms. For substances with very weak bonds, such as molecular dibromine or diiodine, mere exposure to visible light is enough to trigger formation of a fair amount of isolated atoms, something which is exploited in several reactions.
Keeping in mind that isolated atoms are very reactive, they are rather abundant in space. The average density of matter in the Universe is very, very low (about 1 atom per 5 cubic metres of space!), so isolated atoms can go a long time before finding anything to react with. This makes them extremely important for understanding astrochemistry.
