Managing the inventory in Resident Evil 2 is kind of one of the challenges of this survival horror game, but it's possible to increase the inventory space you start off with. In order to increase your inventory space in Resident Evil 2, you need to find an item called the Hip Pouch. This is a consumable item that will permanently increase your inventory space by a little (each gives you 2 additional inventory space slots).
There are a total of 6 Hip Pouches hidden in the game, and you will get a decently sized inventory space if you manage to find all six of them (+12 inventory spaces in total). Their location is the same whether you play as Claire or Leon, but they will be in a slightly different position on your second play-trough of the game.