Just a heads up, spanning 22' while being limited to 4-6" floor depth is extremely cost prohibitive. Deeper is always cheaper, but it sounds like you don't have the space to make this work as planned.
What's below this storage space you want to create? Can you legally reduce the head height by framing the floor lower? This would allow for more storage space as well as room for a deeper and more cost effective floor.
Use one 22' long W8x48 steel beam down the center of your 22'x28' rectangle. This steel beam is 8.5" deep, plus a 1.5" deep wood nailer plate, plus 0.75" floor sheathing, means that the floor is 10.75" at the deepest point. Then use 2x10 joists spaced at 16" on center spanning 14' on each side of the steel beam.
You use top mount hangers connected to the wood nailer plate on top of the steel beam. The Simpson JB210A will work.
If you wanted to save some floor depth, you can skip the 1.5" deep wood nailer plate and field weld the top mount hangers directly to the steel beam's top flange. If you do this, use the Simpson LB210AZ hanger instead. Your floor would now be 10.0" at the deepest point, but in my opinion the extra work of field welding hangers isn't worth the 0.75" of depth that you'd save. (It might seem like I did the math wrong, but your deepest point is now your floor joists, not the steel beam.)
This steel beam's end reaction is up to 9,000 pounds fully loaded, so you will need to verify your foundation can handle it, as well as the columns used to support it.
This is the most cost effective way to do this, but this steel beam alone weighs 1,056 pounds, so you're going to struggle to install it yourself.
This is an expensive option, but you could use six W6x25 steel beams that are 22' long spaced 48" on center and framed with 4' long 2x4 joists spaced at 16" on center between the steel beams. If you field weld top mount hangers, and use 0.5" floor sheathing (32/16 APA span rating), the deepest part of your floor is 6.88". If you are okay with an additional 0.25" of depth, then space your 2x4s at 24" on center and use 0.75" floor sheathing (48/24 APA span rating).
The good part is that these W6x25 steel beams only weigh 550 pounds each, but the bad part is that your steel bill is going to be closer to $5,000 instead of $1,500 for the W8x48 option. I'll let you add up the cost of the lumber, and switching from 2x10s to 2x4s should give you a little bit of savings to offset the additional steel cost.
I'm not sure on your absolute height limitations, so hopefully it's worth the additional cost to save 3.12" to 3.87" of depth.
If you absolutely need the minimum depth, and money is no object, you could also use seventeen 22' long W5x19 steel beams spaced at 19.2" on center, skip the wood floor joists, and install 0.59375" floor sheathing (40/20 APA span rating) without a wood nailer, you would have a 5.75" deep floor. This will cost close to $10,000 in steel, but each steel beam will only weigh 418 pounds so it might be easier to install yourself.
If we change gears and stay away from steel beams, and go with LVLs instead. You could use 22' long 4 ply 1.75" x 7.25" 2.0E LVL spaced at 12" on center. This would still be about $7,500 in LVL, so not much cheaper. It'd be 7.75" deep, but much easier to install because each ply weights only 82 pounds, or about 330 pounds for the 4 ply beam after it's nailed together.
Oh, I assumed you were trying to add this to an existing structure. If the 22x28 structure isn't even built yet, then you have another option! Have the roof truss manufacturer build room in attic trusses or storage trusses to create this storage area up in the roof. 6 foot tall wood triangles are much stronger than 6 inch tall steel beams. You can even specify a deeper heel height and get a little more space by raising the roof and getting more space for insulation as a bonus.