Typically, indie games save to your local device only. These game saves are small though; they probably aren't causing your space issues. Getting an external hard drive might be a good solution for you.
To save games online, we must pay for a server to host the game saves, pay for a login system that is secure (can't be hacked), then code the game saves for the server (probably using PHP). You also need a secure system to reset a password. Every time you switch devices you'd have to login to that account and download your latest data. Because it saves online, you can never play offline - so it's constantly using data.
I had personally investigated that process of saving progress online, and it only becomes viable - time and money wise - if you have thousands of players. None of my games have more than a handful of players.