Your turnouts, boots, mask, hood, gloves, helmet and other little hand tools and whatnot should fit comfortable in a large lidded bin that you can purchase at any home-goods store or the like. Now if you're hauling around your own NY Hook, Pike Pole, Axe or even Halligan type bar, then you may have storage issues.
Walmart actually has a decent sized large plastic storage bin - heavy duty - that's lockable which would sit under the bed rails. I know because I recently purchased one for my garage and it sat in my bed very nicely on the way home [HERE
]. It's not the most visually appealing piece you'll find [ugly] but seeing as you have a bed lid and functionality is your primary concern, it may or may not fit your needs....
It will definitely have enough room for the primary items mentioned earlier and then some more.
I know what we carry in our engine's jump bag but as EMT's, we certainly don't have the equipment that a Paramedic would [should] carry. However, I can't imaging a Medic's bag carrying much more supplies than we do as on scene, a basic jump bag is all that's needed until patient's are in the back of the box, stabilized and being transported, right?
Hope this helped some and thanks for you do.
Good luck to you.