Is an unsigned work contract binding?

The contract itself may be uneforceable, however the contractor can still take legal action to sue you under other causes of action than Breach of Contract.  He can sue for Unjust Enrichment and Promissory Estoppel and maybe Fraud.  The contract can be used to show the intent of the parties possibly depending on what other correspondence exists. You should see if you can come to an accord with the contractor and pay for what has been done.