Can you legally enter into a contract on Saturday or Sunday?

On Saturday, yes.

Sunday is a different matter -- Common Law [Common Law is old English law based on custom rather than a formal statute.] forbid making contracts on Sunday, and if done were illegal. In the absence of statutory law specifying, some judges honor common law and some don't. Some states forbid them, most don't.

Find out if your state has a statute that defines it. If not, they will probably be legal but if it's easy to avoid Sunday I'd do it.