How does an ordinary object become radioactive? In the 2019 miniseries "Chernobyl", ordinary objects are depicted as being capable of becoming radioactive, such as clothes, water, stones.
How exactly does something composed of a non-radioactive mass, become radioactive?
I'm aware of the differences between alpha, beta, and gamma radiation, and I know how ionizing radiation works.
However, it isn't clear to me how any radiation, including ionizing radiation makes something radioactive in any long-lasting sense of the word.
I can imagine that ionizing radiation excites the atoms in the object, which makes the atom emit a photon until it becomes relaxed again. However, this doesn't sound like something that has a very long lasting effect?
I can also imagine that radioactive particles, such as those from U-235, may stick to clothes or contaminate water. However, this too seems not that plausible, is there really that much U-235 in a nuclear reactor for dust particles to be a considerable problem in this regard?
I'm not arguing that this isn't true, it simply isn't clear to me how the mechanism behind it works. I'm pretty sure this isn't clear to most non-physicists either.
 A: I ran a measurement lab in Surrey during the Chernobyl crisis and carried out whole-body measurements of many people, including bus-loads of school children returning from the general area.
What I detected was primarily "U235 fission fragments" (Google the quotes) which are the unequal sized 'halves' of 235 - lots of mass numbers around 90-100, lots around 130-140. The people that were contaminated had been caught in the rain or walked in puddles. The rain took particles into their hair which lodged in the microtexture of the hairs themselves. These don't wash out easily and had to be cut out. The nature of the particles suggested that they were smoke from a very intense fire that was able to volatilise the normally refractory isotopes of Cerium and similar. When these cooled, they picked up other nuclides including I131 and Cs134 and Cs137. Whole body counts after showering and radioactive hair removal detected very low levels of thyroid uptake.
Later I had the opportunity to count the levels on MSC filtration systems. This was interesting but I never got a satisfactory answer. The filters are 3 stage, prefilter that removes coarse dust from the air, a fine filter that takes out most of the rest and a HEPA (High Efficiency Particle Absorption). The main nuclides that I was looking at were Iodine and the Caesiums. I found that the ratios of Cs134 to Cs137 were different between the different filters, but consistent in different parts of the same filter and between filters of the same type. I could only conclude that the different particle sizes came from fires of different intensities in different parts of the reactor. These different parts may have had fuel rods of different ages explaining the different Cs ratios.
I was also asked to count the contamination levels on samples of herbs and spices imported by a friend of my Professor from different countries that had different exposures to the radioactive clouds. Sage in particular was contaminated, presumably because the furry leaves collected particles in a similar way to human hair.
The Caesium issue was thought to be a biologically transient one. It was expected that any that was absorbed would be flushed out of the body along with potassium (sodium has a specific mechanism to reabsorb it if required). This was found not to be correct and Wales had a long-lasting crisis because the caesium was remaining in the soil, reappearing in the grass and being taken up in sheep. Some farms were not allowed to sell their sheep for years.
Gamma-neutron reactions require a gamma energy in excess of 6MeV - rare in reactors.
Fission continues in the reactor to this day. U235 is spontaneously fissile, and alpha particles from the fuel rods will generate neutrons if they interact with low-Z elements. These neutrons will stimulate fission in turn.
Neutron activation is a problem. Gold jewellery or fillings are not permitted in staff in a nuclear facility because the cross-section fot gold is so high. Other substances are also easy to activate - steel invariably contains other transition metals, especially Cobalt. That becomes activated into Co60 in even a mild neutron flux. Fortunately not particularly fast.
A: There are three main effects:
The first, and simplest, is particulate contamination. The uranium fuel rods were pulverized in the explosion and so dust particles contaminated with uranium and other isotopes (fission products in the fuel rods) were scattered to the wind. Don't underestimate the amount of dust and smoke released. There were several tons of highly radioactive material in the core. That makes a lot of dust.
This is what produced the long-travelling dust cloud that triggered detectors in Minsk and Sweden and elsewhere. The dust can get on clothes and can be transferred by touch in the same way any contamination is spread. The problem for health is that each tiny dust particle contains trillions of radioactive atoms that are constantly decaying and emitting radiation. If you get some particles in your lungs, they will sit there radiating away into the surrounding tissues for many years. Not good.
The second effect is from immediate (prompt) gamma radiation from the core. This  was what produced the light effect above the reactor and why Legasov wouldn't let the helicopter pilot fly over the core. This is mainly what killed the firemen and the shift crew. Here, you have basically a beam of radiation coming directly from the dense core and the immense rate of decay occurring there.
A third effect, is that the intense radiation (gammas and neutrons) can affect nuclei in stable atoms and activate them. That is, it converts the stable isotopes into radioactive isotopes, which will later decay. This is well-described in the other answers.
A: As far as contamination by radioactive particles goes Uranium-235 is  not very radioactive, in the sense that it has a half-life of 703 800 000 years. Caesium 137, one common rest product from fission reactors, has a halflife of 30.17 years. This means that if you measure radioactivity as number of decays per time unit Caesium 137 is  $703 800 000/30.17 \approx 23 300 000$ times  more radioactive than the same amount of uranium.
As per regards to small amounts of dust particles, the Avogadro constant tells us that one mole of a certain atom or molecule contains $6\times10^{23}$ such objects. Now as we know that Caesium 137 has a halflife of 30.17 years we know that if we have one mole (136.9 gram) of caesium 137, $3\times10^{23}$ of those atoms will decay in 30.17 years. As 30.17 years contains: $30.17\times365\times24\times60\times60=951 441 120$ seconds we find the average number of decays of one mole of Caesium 137 per second during a period of 30.17 years to be $3\times10^{23}/9.51441120\times10^8\approx 3.15\times10^{14}$ decays per second. So very small amounts of a radioactive substance generate a lot of decays per second.
If you formally want to know "How exactly does something composed of a non-radioactive mass, become radioactive?" besides from becoming contaminated by a radioactive substance, you can go to wikipedia and look up  induced radioactivity. It basically says that the main mechanism behind induced radioactivty is capture of neutrons by a previously not radioactive nucleus. So if you have a non-radioactive substance next to a radioactive substance that emits free neutrons the non-radioactive substance might catch some neutrons and become radioactive if it now is an unstable isotope.
There is another form of induced radioactivity when you hit a nucleous with a gamma ray with enough energy to free one of its neutrons. If the new isotope, with one neutron less, is not stable you have induced radioactivity.
A: 
I can also imagine that radioactive particles, such as those from U-235, may stick to clothes or contaminate water. However, this too seems not that plausible, is there really that much U-235 in a nuclear reactor for dust particles to be a considerable problem in this regard?

Yes, there's really that much radioactive stuff in the reactor.  However the problem isn't the uranium.  When the reactor starts operating, it produces decay products which are much "hotter" than the uranium.  They're so radioactive that even microscopic amounts are dangerous.
Other materials can become radioactive, but only if under intense radiation (such as the inside of a reactor).  The "contagion" shown isn't from induced radiation, but from transfer of particles containing hot decay products from the reactor core.
A: Clothes, water, soil, etc become radioactive because they get contaminated with fission products like cesium-137, iodine-131 etc.
