Yes, three people can play Chinese checkers, and the setup is the standard one with half the triangles left empty. Here is the whole thing.
Setup for 3 players
The board is a six-pointed star with 121 holes: a hexagon of 61 in the middle, and six triangles of 10 at the points.
With three players, take every other triangle, so the three of you sit evenly spaced around the board. Each player fills their triangle with 10 marbles of one colour. The three triangles you skipped stay empty for the whole game.
Your goal is the empty triangle directly across the board from you. That is the part worth noticing: with two players you race into a triangle your opponent is sitting in, and with three nobody starts in anyone’s destination. Every player has a clear runway from the first move.
How the marbles move
You move one marble per turn, and there are two kinds of move.
Step. Move a marble into an adjacent empty hole. That ends your turn.
Hop. Jump a marble over a single adjacent marble of any colour, landing in the empty hole immediately beyond it in a straight line. If another hop is available from the hole you land in, you may take it in the same turn, changing direction as often as you like. A long chain of hops can cross the whole board in one move.
You cannot combine a step and a hop in the same turn. Nothing is ever captured, and no marble is ever removed from the board. Everything on the board, yours and everyone else’s, is a stepping stone.
How to win
The first player to fill all 10 holes of their destination triangle wins. Most groups keep playing for second and third rather than stopping there.
The blocking problem
Standard rules say nothing special about a marble sitting in someone else’s destination. If an opponent is parked in the triangle you are trying to fill, you wait for them to clear out.
Most groups find that unsatisfying and add a house rule, usually that the blocker must move on request. There is no official version, so settle it before the first move rather than during the argument.
Standard rules are also silent on whether a marble may stop and rest in a triangle that is neither its start nor its destination. Hopping through is universally fine. Stopping is worth agreeing on too.
A variant worth knowing
Three players can also take two sets each, in opposite triangles, filling all six points of the star with 60 marbles. Each player races both colours across. It is a longer, denser game with no empty triangles, and it is the usual answer when three people want the board fully used.
Why three players is a different game
With two players the game is zero-sum: any position is a win, a loss, or a draw, and that is the whole story. Three players break that. Once a player can no longer win, every move is equally worthless to them, but the move can still decide which of the other two wins.
That is kingmaking, and it is not a quirk of casual play. I solved the smallest nontrivial three-player Chinese checkers board exactly, and in that solve every reachable position for the first eight plies is a kingmaker position. No player can force their own win, any two can force the third out, and all six ordered kingmaker relations are live from the opening.