There are two questions here.

Q - Can a child travel alone by train?

A - Every train company sets their own rules for what age a child can travel alone. I know that my local train company doesn’t set an age. Just that the child must be ‘mature enough’. But you need to ask every train company you will use.

Q - Can I buy and use a child Eurail pass without an adult?

A - No. You can only get a child rail pass for free when you buy an adult pass. And the child pass can only be used when the adult is travelling. But the child can maybe buy a Youth or Adult pass and travel alone - on train companies which allow it.

There can be problems for a young child to cross borders without a parent/guardian so you need to check the rules for the countries concerned if there is a border crossing involved.