Online Modded Minecraft Server with a Mac?

I want to make a modded server for Minecraft with my Mac to play with my brother. I've already done something like this, but it was just a world on my Mac, where he had to connect through LAN. I would like it to be an online private server (if you've ever Pixelcore, this'll be familiar) that doesn't rest on any computer so both he and I can both access it without the need of the other one opening up his Minecraft.

Added (1). Just to clear it up: I want this world to be essentially in a cloud, where it doesn't need to rest within a computer, and we can both access it without the other.

You would have to hire a server on the Internet. And very few hosting companies will allow you to host games as they use masses of server resources. You need to have a computer running the server program. Get an old machine which is no longer good for rnning Windows, and install a Linux distribution, then install a Linux version of Minecraft.

A computer somewhere is essential to running a minecraft server. It is not just accessing a file somewhere, it needs to handle the user connections, generating the world as users explore it, and creating mobs in survival mode. There are places that rent minecraft servers that you can control, but for just the 2 of you, it could be run on any old computer you have or can get. I have run a PC minecraft server in Linux on as little as a 1 GHz 2 core tablet PC with 2 GB RAM. So even an old laptop would work (it does not need good graphics to just run as server). You just need to know how to port forward on whatever router you have and the person running their client locally likely would need to connect to the local LAN IP of the server instead of its public IP. The only computer that would need to remain running (set to NOT go to sleep) would be the server.

It is possible to add a single player world to a server, but if you add mods, new things from the mods may only show up in newly generated areas not previously explored by anyone.