So I have my main computer (Windows 10 (it s amazing)) and I want to have a minecraft server. But I don t want to have it saved on this computer because I don t want it to run all the time. So my question is the following: Can I buy a small computer and plugin external disc into it and save the server on the disc and have the small computer running all the time and while doing all that, is the server able to broadcast from the disc?
In theory, yes. All that you would be doing is turning that small PC into a Minecraft server with an external drive.
Would it be optimal though? no. Minecraft is usually intensive on hard-drives and where it saves it's world data to. Having it run off of an external drive brings along one major issue - the time to read/write data is nearly doubled (or triple) since it will be on a slower USB connection rather then (directly connected) SATA.
If you really want to take this route, have the PC able to support an internal drive of some kind. If it's going to be a "micro" style of PC (and not explicitly built to play MC on it), download and run the server software instead of the full Minecraft client.
'Small computers' may be more efficient but for hosting a server you're probably going to have many frustrated players. As was mentioned, a lot of data is being read from and written to the disk, particularly with multiple players. While it might work, your players would likely experience lag such as delays in mining blocks, frozen mobs, and a long wait when using a portal.
A decent 'normal computer' with an internal drive would likely be a better option for performance. Just stop the Minecraft Server program when you don't want it running and start it again when you do.