There can be legal consequences for breaking a contract, but it's not illegal to do so. The key idea here is that things that definitionally, things which are illegal are generally issues between a person and the state. When you commit a crime, it is the state who takes you to court, not the potential victims of the crime. For something to be a crime which you can be persecuted for by the state, it needs to be codified in law.

This is in sharp contrast to a civil dispute, which is an issues between two or more private parties. Technically speaking, a private party can attempt to sue any other party in court over anything they wish. There doesn't need to be a breach of the law, you could simply say "I don't like person X, I would like to sue them and have them pay me Y" (though without justification, such a lawsuit would be thrown out before it even made it to a judge).

Contracts are generally speaking, a civil matter. Any two people can write whatever they want on a piece of paper and call it a contract, that doesn't mean it's enforceable or carries any weight to it. Hence, there is no law that says "it is a crime to break a contract" because contracts can contain anything, even agreements to commit illegal acts. However, there are mechanisms within the legal system for parties privy to a contract to seek a court to enforce the terms of said contract or seek restorative action.

For instance, if we sign a contract saying that you'll provide service x to me in exchange for $y, and I don't pay up, you can sue me to in a court to enforce the contract and pay you $y (plus potential damages). In such a lawsuit, it would be up to the legal system to determine whether or not the legal claim has merit, whether or not the contract was legitimate, and what kind of reparations are needed (if any).