Never mind contracts. A society without money is one where civil law becomes worthless.
To take an extreme:
Mary is walking along the streets with her kids. Mary gets hit by Paul's vehicle as Paul is in a crosswalk.
4 days later, Mary dies. The police determine it to be an accident, so no criminal charges are filed.
Joe, Mary's husband, would (among other causes of action) have the right to sue for wrongful death. Whether or not you believe society is too litigous, the fact is he has a right to sue.
But in order to file suit, you have to have a remedy the court can impose. It's not a criminal case (it's a historical artifact that the idea of a private person bringing a criminal case even needs to be mentioned - only the government, IRL and certainly in any "future" criminal justice system, would be able to bring criminal charges, for a whole host of good reasons), so Paul isn't at risk of imprisonment or anything similar.
In civil cases in real life, you have broad classes of remedy you can seek for any civil wrong done: Monetary damages, specific performance (only, really, in construction/real estate cases; as Myasischev mentions, any other time it looks a little too much like servitude), or injunction [a court orders you not to do something].
Wrongful death is a great cause of action to use for an example because it slams you right up against the heart of the matter.
Of the three categories of remedy, let's look at each one in the context of a wrongful death case:
Specific performance: What the heck is the court going to order you to do? Not at issue.
Injunction: Killing someone is a cause of action anyway. You're generally not supposed to kill people, anyway, and the death has happened, so how is this a remedy? How does it "make the claimant whole"?
Monetary damages: Wait, scratch that. You can't sue for monetary damages in the Federation/on Earth, there's no money to sue for.
In law, every right must have a remedy. Take away monetary damages, you take away a huge reason for people to, say, exercise care when operating a vehicle. More importantly, you make it impossible for someone against whom a wrong has been done to be made whole.
How the hell is there justice in the Federation, then?