A proton that is not bound into an atomic nucleus is stable (the mean lifetime of a proton which is at least 1.29*10^34 years, which implies that it would happen in not more than 1 in 10^24 protons over the entire 1.38*10^10 year lifetime of the universe), a result which follows naturally in the Standard Model from conservation of baryon number, quark confinement, and the fact that the proton is the baryon with the lowest rest mass.
When a neutron is not bound into an atomic nucleus and is at rest, it has a mean lifetime of about 14 minutes and 42 seconds +/- 1.5 seconds (which is equivalent to a half-life of 10 minutes and 11 seconds +/- 1.0 seconds), and naturally decays via weak force into a proton, an electron, an electron anti-neutrino, each of which has kinetic energy (and in about one in 1000 cases, also into electromagnetic energy in the form of a photon). The kinetic energy and the photon energy combined are equal to the difference in rest mass between the neutron and combined rest mass of the proton, electron and electron anti-neutrino times the speed of light squared, which turns out to be 0.782343 MeV/c2.
This mean lifetime is incredibly long compared to all other unstable particles in physics (except oscillating neutrinos). No other baryon (except the proton) has a mean lifetime of more than 10-10 seconds. No other meson has a mean lifetime of more than 10-8 seconds (the charge pion). The mean lifetimes of the muon (10-6 seconds), the tau lepton (10-13 seconds), and the even shorter mean lifetimes of unhadronized top quarks, the W boson, the Z boson and the Higgs boson, all of which are much less than 10-20 seconds. Gluons don't have a fixed lifetime, per se, but they are typically exchanged between other confined color charged particles in similar time frames, because they travel at speeds approximating the speed of light (a bit less than 3*10^8 meters) over distances on the order of the proton and neutron charge radius (i.e. about 0.8*10-15 meters), which is shorter than the mean muon life time.
The process by which free neutrons normally decay, called beta decay, is what causes nuclear radiation, although the rates of beta decay are lower when neutrons are bound in an atomic nucleus with protons, but the number of neutrons is high in relation to the number of protons.
In contrast, when protons are bound with a number of neutrons that produce a stable atomic isotype, beta decay does not occur and neutrons are stable. In part, this is because the decay of neutrons in a stable nucleus is offset in part by "inverse beta decay" in which an energetic proton emits a neutron, a positron (i.e. anti-electron) and an electron neutrino, and in part by electron capture in which an energetic proton and electron (perhaps one created in the ordinary decay of neutrons in the same atomic nucleus) merge to form a neutron and an electron neutrino (which is the anti-matter equivalent of the electron anti-neutrino produced in ordinary beta decay). There are also lower order (i.e. much less frequent) possible paths by which energetic neutrons and protons can form other kinds of hadrons with or without leptonic decay products.
Empirically, the neutron to proton ratio needed to make an atomic nucleus stable is between 1 and 1.537 (gradually increasing with larger atomic numbers), except in the degenerate cases of hydrogen (a bare proton without a neutron), and helium-3 (two protons and a neutron). As Wikipedia explains:
Neutron-proton ratio (N/Z ratio or nuclear ratio) is the ratio of the number of neutrons to protons in an atomic nucleus. The ratio generally increases with increasing atomic numbers due to increasing nuclear charge due to repulsive forces of protons. Light elements, up to calcium (Z = 20), have stable isotopes with N/Z ratio of one except for beryllium (N/Z ratio=1.25), and every element with odd proton numbers from fluorine to potassium. Hydrogen-1 (N/Z ratio=0) and helium-3 (N/Z ratio=0.5) are the only stable isotopes with neutron–proton ratio under one. Uranium-238 and plutonium-244 have the highest N/Z ratios of any primordial nuclide at 1.587 and 1.596, respectively, while lead-208 has the highest N/Z ratio of any known stable isotope at 1.537.
One can imagine an atomic nucleus which has zero protons and two neutrons, which is called a "bound dineutron". An atomic nucleus with no protons and an arbitrary number of neutrons in known, in general, as neutronium. A bound dineutron particle, like a neutron, would have no electrons and would therefore be chemically inert. It would have a rest mass of about 1.879 GeV before adjusting for mass due to the nuclear binding energy of the two neutrons. Since the strong nuclear force and weak nuclear force only act at short ranges, it would be collisionless and only influenced by gravity at ranges of less than the radius of a typical helium atom nucleus. Thus, if it were stable, it would be an excellent cold dark matter candidate.
Dineutron states were observed in 2012, but were transitory states that were not quite bound, rather than constituting a particle made up of bound neutrons which could in principle be stable. (Incidentally, unlike some neutral mesons, neutrons do not oscillate between neutrons and their anti-matter counterparts, which follows quite naturally from the conservation of baryon number in the Standard Model.) There appears to be a modest shortfall of nuclear binding force between a bound atomic nucleus and the bineutron state, but one could imagine that there could be some special factor that is ignored in other circumstances but is material in a nearly trivial two neutron system. For example, at the high energy scales present in Big Bang nucleosyntheis, the running of the Standard Model parameters with energy scale might permit the existence of bound bineutron states. Similarly lattice QCD studies with higher than physical pion masses (also here) find that bound bineutrons can exist. And other studies can't definitively rule out their existence and they might even play a role in determining alpha decay rates (although experimental evidence pretty clearly rules out truly stable dineutrons).
There also appear to be a set of phenomena common to not quite bound systems of baryons.