Is it possible to store Minecraft not on a computer?
Solution 1:
It might be possible to run it from USB-drive or so, but as stated above, that could cause some issues. I don't think that most of the game data will be stored anywhere other than in your PC, since most of the data goes directly to the %appdata%
directory.
Some third party applications however could be able to change that default saving directory.
Solution 2:
You may run into issues trying that, but you can. In the minecraft executable, before you login you can click Edit Profile to change some settings. You will need to run the minecraft.exe file from the external drive, because it also counts as the launcher.
When you edit the profile you are using to login (or create one for a new profile), you will see an option for Game Directory that lets you change the directory that minecraft installs to. This could be a USB, but I don't know if it will work well with cloud based drives.
Minecraft doesn't take a lot of space up, but that is something you can try.
Solution 3:
You can run it off an USB drive but do be aware that Minecraft will still use your hard drive to save the worlds. You won't be saving a lot of disk space.