You manage your inventory by selling stuff you don't need right now. When you do need it go out and harvest more.
You really can't skill up all classes at the same time, your inventory would be completely screwed. You talk about materials, but what about all the different gear sets needed? Gear changes every 5-10 levels so assuming every piece of gear hgets replaced every 10 levels starting at level 10....that's 55 inventory slots just for gear.

Sell stuff you don't need right or put it into the FC chest, and then take what you need out later. Stuff isn't THAT expensive that you need to hoard everything, if you have no idea when you'll actually use that material or item, sell it and buy it back or harvest more when you need it.