Minecraft Inventory Saving

I've got a solution, but it's a bit ugly; it involves killing the player and having them reorganize their inventory at each save. You'll also need keepInventory off.

Whenever you want to save, teleport the player to an enclosed cell with a hopper on the floor and chest below, then kill them; all of their items should go into the chest. If there were already items in the chest from a previous save, you'll need to /setblock the chest to clear it first. /clone this chest with their items to another enclosed cell.

When the player dies, teleport them to the cloned chest and break the chest. All of their items in the cloned chest (that they had when they saved) will fall into their inventory. /clone the chest again form the original one in case they die again. Then you'll just need to teleport the player out of the cell back to where they were when they saved.

In Minecraft 1.14 and later versions, it is possible to copy the player's inventory to a chest without killing the player.

/data modify block <coordinates of chest> Items set from entity <player> Inventory

Note that this only copies the hotbar and the bottom two rows of the player's inventory, since players can carry slightly more stuff than can fit in a chest. If you want to copy the player's entire inventory to a double chest, you need a much more complicated trick.