Reddit mentions of DuPont TY122S Disposable Elastic Wrist, Bootie & Hood White Tyvek Coverall Suit 1414, Large

Not to be nitpicky, but he's not wearing a hazmat suit. Hazmat suits are meant to totally protect the body against any airborn or touched hazardous materials, and are typically made of some material that doesn't let stuff permeate it.

What this dude is wearing is something more like a disposable paint suit.

These are usually made of a cotton based material and feels almost like you're wearing a big body sized paper towel. Aside from using these to, well, paint, people working around sensitive electronics will wear them as well to prevent skin dandruff, hair, and body dust from contaminating the environment or the electronics they are working on. Here's an example.

Why this guy wears one? Probably to ensure his dna isn't left anywhere. He's like the DA's courier, so this suit helps ensure no trace is left behind.

Respiratory protection is what's most important in these cases, as well as something to keep spores/viruses/bateria out of your eyes. A full face respirator with at least N100 cartridges is a good idea. To avoid having to sacrifice old jeans and a shirt, I would personally cover up with a Tyvek suit. They're cheap, disposable, and will keep things off of your skin.

Z

Try a tyvek coverall suit...Technically they're made for painting but pretty much looks like a hazmat suit. A lab where I used to work actually used then as such to prevent exposure to bloodborn pathogens. Just pair it with some rubber gloves and a gas mask/face shield

EDIT something like this would do. I used one to look like a hazmat suit for this photo. they tend to run on the small side. I'm 6' and skinny as a rail and I need an XL for comfortable range of motion

https://www.amazon.com/gp/product/B0008F4HPQ

Oh I agree it wont be safe, I don't think its going to be safe anywhere in the event it happens.

My main concern is that if you do find a shelter that is open and will allow you in the filters and fans most likely will not be working. Which means none of the radioactive dust that is floating in the air is going to get filtered out.

You might be able to find gas/dust masks that can filter out super fine particles (the biggest threat from fallout is inhaling the radioactive dust). I'd also suggest purchasing iodine pills or liquid:
https://www.amazon.com/iOSAT-Potassium-Iodide-Tablets-130/dp/B00006NT3A

Radioactive iodine will be the majority of the radioactive particles that get attached to the dust. Your thyroid is the main user of iodine in your body so they tend to get sent to that location. If you can replace or fill up the iodine receptors in your thyroid before ingesting the radioactive ones then hopefully your body will excrete the radio active particles over time.

There are other radioactive elements that get produced during the explosion like francium, americium and there are med's available to help get rid of those but they are Rx only and usually reserved for military.

a Tyvek zip up suit might not be a bad idea either, this will help limit your skin exposure.
https://www.amazon.com/DuPont-TY122S-Disposable-Elastic-Coverall/dp/B0008F4HPQ/ref=sr_1_3?ie=UTF8&qid=1504501640&sr=8-3&keywords=tyvek+suit

May your journey be safe with this.

http://www.amazon.com/DuPont-TY122S-Disposable-Elastic-Coverall/dp/B0008F4HPQ/ref=sr_1_2?ie=UTF8&qid=1449852068&sr=8-2&keywords=hazmat+suit