Is it possible to transfer funds from Europe to the US without involving banks at all?

If you are implying fiat, then yes, the method would be Hawala. In a Hawala system, assuming you adopt a cash-based transfer, you would pay someone in cash in the country of origin, and in the payout country, your beneficiary would be paid in the equivalent amount in local currency in cash also.

Alternatively, if you wish to send funds (value) across, you can do it via Crypto only.