In the supernatural plane of existence known as The Exhibition, time appears frozen - nothing is moving, falling objects are suspended motionless, the works. In addition to this, while outside visitors and their belongings move as normal, the inertia of native matter is 100x, maybe even 1000x what it normally is, which means that anything of much mass at all is impossible to budge, you can walk on water like it's stone, etc.
You can, with difficulty, move a feather that was floating in the air, and wherever you managed to move it to, it would remain suspended there. If you succeeded in putting the feather in your pocket, it would be very hard to walk away, as it would probably tear the pocket loose from your clothing and remain where it was in space.
There are some very good reasons to want to explore The Exhibition, which otherwise resembles a medieval realm. However, particulates in the air cause big problems. Water droplets from fog, or dust in a dusty room, even pollen, all accumulate on your clothing and hair and in your lungs and increasingly bog you down. This can be fatal! The modern-day solution would be a protective suit like this dry suit. It would still be hard to move through particulate-heavy air (would feel like wading through water or mud), but it would at least all slide off of you so it wouldn't get worse over time.
How can the canny explorers in my setting visit this place as safely as possible? Their base is a medieval fantasy world, though I'll take any technology plausible up through the Renaissance. There's magic in this setting too, but I'm looking for a mostly or entirely non-magical solution.
It doesn't have to be perfect, just better than nothing, and the less cumbersome the better. Lotus leaves might be a good start, but I fear that fashioning them into clothing would be prone to failure (holes made from stitching would tear when you started moving, etc). Maybe they could be layered overlapping on top of an undersuit with some kind of pitch or glue? But if the glue remained wet it would attract particulates, and if the glue hardened, it may become inflexible.
(Although I'm not aiming for hard-science plausible physics here, any tangents about other effects of this high-inertia plane are welcome, though I might ask another question about that.)
Update: Assume that gases aren't subject to this extra-inertia effect, only liquids and solids. So, in the absence of solid particulates, air pressure and breathing work as normal.