SE has said that they won't create the specialized inventory for crafting materials and bait when that space could be general inventory for any use.
With the majority of materials needed for leveling found on vendors, there's little need to store most of them in inventory. Just buy what you need as you need it. If you end up with extra, sell it instead of holding onto it (unless it's a hard to get material you are certain you will be using in the near future). Something sitting unused in your inventory is gil lost. Just make certain to check the marketboard before buying materials for HW/SB recipes. Many times those materials are sold for less by player than the amount the vendor charges (seen on tooltip).
Another way to save space is to use Ishgard Restoration (4) recipes for leveling. There's a relatively small number of different materials needed for each 10-20 levels up to level 80 recipes.