In Diablo 1, there is no storage other than your character inventory. This can be a bit of an issue, particularly as you collect more and more gold, etc.
However, I saw one playthrough presumably done in one session, where they dropped items in the centre of town. Items such as rings and amulets, that could be useful at a later date.
Do items dropped in Diablo 1 ever de-spawn?
It depends on whether you play singleplayer or multiplayer.
In singleplayer, dropped items stay where they are until you defeat Diablo or start a new game with this character (this also applies to the entire dungeon, no levels will repopulate until you start a new game). You can save and load the game, exit and restart the game, doesn’t matter, everything is saved. Once Diablo is defeated however, the open game of your character ends and items disappear when starting a new game. In singleplayer it’s pretty common to keep a “stash” of gold, spellbooks and other useful items on the ground near the town portal area. Just remember to take it with you before you defeat Diablo or start a new game (don’t worry, Diablo doesn’t drop anything special in Diablo 1).
In multiplayer, it works as @Tom describes. A game stays open as long as at least one character in it. Once no players are left, the game is closed and disappears forever. In multiplayer, you usually use a mule character to serve as your stash. This does however require you to play with a friend you can trust not to steal your gold. Alternatively you could also try to keep the game open forever, but that risks a disconnect and closing of the game. Or you login to the same game from two different machines to handle the muling yourself.