Natural or otherwise, it depends on the half-life of the isotope in question.
More strictly accurately, it depends on the half-life, the decay type and decay energy.
Radiactive decay typically involces one or more of electron emission (beta radiation), alpha particle emission (alpha radiation), high-energy photon emission (gamma radiation), and neutron emission.
Ionised particles (alpha and beta emission) cause damage by basically smashing into other atoms, alpha moreso, because an alpha particle consists of a positively charged combination of two protons and two neutrons (some 8000 times more massive than the electron released in beta decay and with twice the electric charge).
The amount of damage such particles can cause is pretty much directly related to the amount of energy they hold when they are emitted, which is specific to the isotope that is decaying, but for alpha particle is usually around 5 MeV.
Gamma radiation depends very much on the energy of the photon in how harmful it can be. They typically cause damage by ionising the atoms they pass by.
Neutrons can hit other nuclei and make them into unstable isotopes. This requires a direct-hit on the nucleus, and the vast majority of every atom is empty space, so neutron flux has to be quite high for this to be hazardous. A single decay has a very low chance of being harmful, compared to the potential of an alpha particle. Neutrons, however, being uncharged, can travel a lot further, so the safe distance from a neutron source is a lot more than the safe distance from an alpha source - alpha particles typically don't get more than a few cm, but neutrons will travel many metres. Free neutrons also have a half-life of around fifteen minutes before decaying into ionised particles (a proton and electron) and a neutrino. If it does so when it's inside you, that proton and electron are going to cause damage to whatever is around them.