I just finished reading Adam Higginbotham Midnight in Chernobyl, but perhaps more importantly I'm a nuclear physicist who works in radiation instrumentation, so I'll take a stab at this from both the history and the physics angles. Higginbotham backs up the show's account that men were largely sent in with very little protection, usually a rubber/plastic body covering, a respirator (I don't believe they were gas masks, the term Higginbotham uses repeatedly is "petal respirator" which appears to be something like an N95 mask, though of course the filtration capabilities will depend on the specific kind), and perhaps some sort of lead covering. Your question has two parts to it, as far as I can see:

1. Why didn't they use Hazmat suits?

2. Why did they go in the way that they did?

I can't speak specifically to the Hazmat issue, as you'd likely need to know the exact history behind hazardous waste cleanup in the USSR to know what their suits were like, but the main issue is that even modern Hazmat suits wouldn't be much use. To understand why, you need to know a bit more about radiation.

Laymen tend to think of radiation as one big, scary thing, but really there are multiple types and the way they interact with the body is different:

1. Alpha. This is a He nucleus ejected from the isotope's nucleus. They are very energetic and very damaging, but they're also not very penetrating, being easily stopped by a thin layer of material (including skin). If you get an intense alpha emitter inside you it's really bad, but externally they're mostly harmless.

2. Beta. This is an electron ejected from the isotope's nucleus. Electrons are small, so this can penetrate through material, exactly how far will vary based on the material and the energy of the beta particle, but usually it'll take <1cm of lead.

3. Gamma. This is an extremely energetic photon emitted from the nucleus of the isotope. Gammas are very penetrating, but also somewhat less ionizing. It can take many cms of lead to completely stop them.

Also, keep in mind that absorption of radiation is a quantum mechanical process, so there's not really any point at which you can say definitively that all the radiation is stopped. It will always depend on the intensity of the source.

The other important thing to know about radiation is that your body can take it. Everything is radioactive at some level, and you're experiencing it all the time. So there's some amount that you can take and expect to survive relatively unharmed (though this is somewhat contentious, the prevailing model says that any radiation exposure has some chance of harm, even if it's small, but some people believe that at low levels radiation can actually be positive). And what you care about is called the dose, which is the total energy deposited by the radiation (modulo some coefficients for various organs, some are more radiation-tolerant than others). Dose is going to be equal to rate times time. So you can either try to reduce your exposure rate (shielding), or reduce your exposure time.

So, back to Chernobyl. The problem they were trying to solve was the graphite was extremely radioactive. They attempted to use robots to solve the problem but they all failed in the intense radiation fields. The exact strengths of the radiation fields varied wildly around the site, and with them the amount of protection you'd need, although the sites contaminated with graphite were so radioactive with beta and gamma emitters that the shielding required would have made movement so slow as to be more or less impossible.

Given that the body can take some radiation, the decision was made that it was better to use thousands and thousands of men (the so-called "liquidators"), each with only a few seconds or minutes of exposure, depending on their job. Given this, and what I described above about dose, the priorities were speed, and keeping radioactive dust off of the body and lungs.

So, perhaps now you see why they did what they did. Minimizing the time exposed while taking precautions to ensure that there was no prolonged exposure after leaving the scene was actually the best option. The scientists investigating the accident, trying to find the missing nuclear fuel, and trying to prevent the worst-case scenarios (second meltdown, "China Syndrome", etc) also used these principles, shedding shielding and opting for speed instead.

Hopefully this answers your question and is up to snuff here. Main source for history is Midnight in Chernobyl, the general radiation science stuff you could find in Knoll's Radiation Detection and Measurement, or various other sources (this is pretty well established science).