The best solution I found to this is to use fast travel to travel to the nearest thing you've discovered. Or perhaps the second nearest. The point is to use it as a way to pass time, but not to make you travel far away from what you're doing.
So far for me, that makes the storm pass and no rads are absorbed.
Per your comment about being hit by these storms in gun fights, this won't help in those cases since you need to be a little ways away from hostiles in order to fast travel. But if you can break off from the fight (become hidden) or end it quickly, the fast travel will still be of use.
Also, as kalina mentioned in their answer, a radiation suit will help. There's a free Hazmat Suit with 1000 Radiation resistance in the Cambridge Polymer Lab. Here's the CPL's location:
I didn't think to take a screenshot of the Hazmat suit's specific location. The Hazmat Suit itself is on the first floor, in a room across from Ericka's lab(where you mix ingredients plus an isotope), and sits on top of a shelf in the corner. Below is where I think the hazmat suit was, but again I forgot to screenshot before picking it up.
Having such a suit will allow you to do some exploring in the open world during a rad storm, but as mentioned in the other answer it comes at the cost of your armor stats.
The following crafted items provide rad resistance:
- Cooking: Baked Bloatfly (+25), Vegetable Soup (+15)
- Chemistry: Glowing Bloodpack (+75)
Mutant Hound Chops reduces the current radiation level by 50 and can be crafted from Mutant Hound Meat, which is relatively easy to come by in comparison to frequent RadAway.
Best Answer
It is unclear whether standing in irradiated water in vanilla Fallout 4 survival mode causes illnesses in particular:
The FO4 Wikia only notes that
increases the risk of contracting an illness.
In addition, a Steam thread comment describes how
This claim is repeated here (by the same user).
Therefor, I assume that the contraction of diseases by standing in water is likely (but not certainly) implemented by one of your mods.
Even so, wearing a hazmat suit, or consuming Rad-X, should be able to influence it, since both reduce your rads or rad absorption, making you less prone to contamination. Whether they can negate the direct effects on standing in water, depends on the radiation levels of the water and the duration. The Ghoulish perk won't decrease the amount of radiation you absorb altogether, but will return a percentage of it as health.
I suggest you look here for additional information on gaining radiation resistance, and here for an overview of its (percentual) mechanism.
Note as well, that Fallout 4 most likely has a Rad Resistance cap, like (some of) its predecessors:
* https://fallout.fandom.com/wiki/Condition#Illness
✝ https://steamcommunity.com/app/377160/discussions/0/2906376154334607210/#c2906376154334744788
○ https://fallout.fandom.com/wiki/Radiation_Resistance#Notes