Sharing a world with family in Minecraft

I'm thinking about getting Minecraft for a few family members. I know it's possible to play together using the LAN option, but I'd like for my family members to be able to load up the world and work on their stuff even if I'm not playing at the moment. I don't want to use the realms, however. I found an article showing how to make your world accessible from multiple computers by essentially moving the saves location to Dropbox. I can't see why this wouldn't work for multiple people's computers, as opposed to multiple computers owned by one person unless the worlds are directly linked to the creator's account or some such.

Using this method, if my family members all had their Minecraft saves set to save to the same Dropbox folder, would anyone be able to access and play on anyone else's world, even when the person who originally created the world isn't playing? And would multiplayer still work properly when we wanted to play simultaneously?

The setup you describe is a recipe for getting your world corrupted as soon as two games try to access it simultaneously. The single player mode simulates the world on its own - when you start two single-player games from the same world files, you'll have two separate parallel divergent simulations of that world. The worse thing is that the game assumes it has complete exclusive access to the world files and writes to them at will. If two games try to write upon the same files there will be conflicts and the world might end up in a corrupted state.

This is exactly the reason that you need a Minecraft server for multiplayer - the server simulates the world and writes to world files, while the clients just relay your actions to the server and show you the result of the simulation from the server. The easiest way to setup a Minecraft server is indeed Minecraft Realms, but you can setup your own server as well: https://minecraft.gamepedia.com/Tutorials/Setting_up_a_server The server needs to be installed on a machine that is accessible from all clients and needs to be running whenever someone wants to play (in most cases you'd just leave it running at all times).

Additional note: When you open your world to LAN, your game is the server that simulates the world and saves files and the joining player game is a client.