Is Chess with more than three players exists or would be playable?
Yes, chessboards for 3 players exists. I have a similar to the following at home:

https://www.chesshouse.com/products/3-player-medium-wood-chess-set

It is playable and it is fun, however don't expect a really deep chess game. The main tactic is to be not the one who gets ganged upon. The better chess player won't win. The player who can manipulate the best will win (by making the other one vulnerable to attacks or just by talking). 

It is almost impossible to defend against two opponents. For example one captures a piece and the second player captures another piece that was defending the first piece. Another situation is as one attacks your king ('check') and the other one attacks your queen. There are also some funny checkmates, like Scholar's mate where one person uses the bishop and the other uses the queen. 

Be prepared for these situations, otherwise it feels very unfair. By times very funny, but also very unfair :)