What do I do in a Radiation Storm?

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.

Radiation storms are not all that dangerous. At most, I usually soak up about 60-100 RADs. Easily curable with a paltry 40 caps at a settlement doctor. If you are earlier in the game, the best things to do are:

1. Hop in your power armor and wait it out
   Whether it has a charged fusion core or not, any power armor will add hundreds to your RAD Resistance, which is more than enough to completely negate the minor bursts of radiation from a storm. Most storms pass in about 60 seconds or so, so just buckle up and wait it out. Fusion core power is only drained by movement and combat actions, so if you sit still and refrain from doing anything, you can sit through the storm without depleting your valuable cores.

2. Hide in an "out-map" area
   An out-map area is any area that requires you to load out of the "Commonwealth" over-map. Structure interiors, vaults, certain caves, etc. There are very few exceptions, such as areas that require a load, where you remain outdoors. The downside of out-map areas is that you can not tell if the storm has passed. Storms can vary in duration, and are fairly frequent. You may end up waiting, only to emerge into an entirely new storm, but this is not a common occurrence.

3. Carry a radiation suit with you everywhere
   Similar to power armor, a radiation suit offers an inordinate degree of RAD resistance, and can shield you from a radiation storm. However, carrying it with you always eats up precious inventory weight, that might be better spent carrying junk. It might also be a hassle to rush to equip the suit, only to re-equip your original gear about 60 seconds later. This becomes especially cumbersome when you use pieced armor such as raider, leather or metal.

4. Fast Travel
   Arguably the easiest solution. Simply fast travel to a nearby location, then back again, and the storm will have passed. If that spot happens to be in the middle of nowhere, try placing a map marker, before your first fast travel. This will allow you to hone in on your exact previous location. Other downsides to this include spawning into a repopulated area, which means fighting on arrival. Also, if there are corpses you have not looted when you travel away, they may despawn by the time you return.

You could wear a radiation suit but it would come at a penalty of all of the stats you get from your armor. Wearing a radiation suit would effectively protect yourself from any radiation.

It is also possible to gain radiation resistance from armor which would reduce the amount of RADs gained. For temporary resistance to radiation you can also take Rad-X.

The collectable book Astoundingly Awesome 10 (available from the church near Trinity Plaza) adds a +5 radiation resistance.