Surprisingly, storing a game (with all moves) can take less space than encoding a single board. This is because you can effectively encode a move in a single byte (as there are less than 255 moves possible in each position). Applying compression to the resulting binary string will allow you to reduce the space even more.
And shameless plug: Using this encoding, I'm storing millions of games on https://www.chessmonitor.com/ There you can link your Chess.com or Lichess account and view all kind of statistics of your games.
Check out this great blog post of Lichess for more information: https://lichess.org/blog/Wqa7GiAAAOIpBLoY/developer-update-2...
And shameless plug: Using this encoding, I'm storing millions of games on https://www.chessmonitor.com/ There you can link your Chess.com or Lichess account and view all kind of statistics of your games.