RE5's inventory system was designed to go hand in hand with the co-op aspect, you aren't supposed to horde items like RE4, the game gives you just what is needed, if you could carry more, the game would probably be way too easy. You can't carry all the armor (Bulletproof vest and melee armor), weapons, and herbs, but rather enough room for, say, an herb and some weapons, or armor and weapons.Fortunately the herbs and first aid sprays are plentiful in this game so I never had a problem with that. Enemies always seem to drop a health item when you are low on health anyways.
One thing I did find annoying was the inventory system. It's like they took a step backwards from the improved inventory system of Resident Evil 4 and gave you very limited space to work with making it frustrating to manage all your items (especially the ammo). They should have at least given infinite capacity for storing ammo in one slot so that it doesn't quickly take up all the inventory space.
Also Sheva and Chris don't help anything by constantly handing you useless items.