Is it possible to get married without a marriage license, at a church, but be married in God's eyes?

God does not work by issuing licenses. A license is a political thing, a pretense that you need permission from the local government. Marriage is a natural condition: you are married if you DECIDE you are married. No license is needed and no politician has power to forbid it. There is also a three way corporation involving a man, a woman, and the state for the purpose of making babies, and that corporation is also called marriage. The license was invented after the civil war specifically so it could be refused to mixed couples. Of course that meant mixed black and white, since nobody cared about any other combination.

If you decide you are married but don't tell anybody, that is called "shacking up". It is politely called common law marriage and local laws may or may not recognize it. But a record of marriage in a family bible is acceptable evidence of marriage in any court in the country. If you get a marriage license, you have given your children to the state. After all, the state is one third owner of the corporation that produced them!