YES, the technology is feasible for the time.
Romans used hamster Wheels to build the Coliseum: See this clip from the Science Channel of one in action:
One girl can use it to lift a 1200 lb block, and three students together raise a 5200 lb block off the ground. There is a drawing of such a wheel from Roman times; and the wheel and the compound pulley it uses were known to Romans (although the compound pulley may not be necessary for powering a ship).
Other examples are shown here, as cranes, plus examples proving the Romans knw how to use various forms of gearing.
Such a wheel is almost entirely powered by legs. Romans were very familiar with gearing as well; and were making both wrought iron and steel in the Noricum province (https://en.wikipedia.org/wiki/Roman_metallurgy).
The Hamster wheel creates power in a rotating axle; using simple gearing it could be used to turn a propeller. Note also, sealant is not necessarily a problem, the equivalent of an outboard motor propeller is easy enough to engineer: An outboard motor uses a propeller without piercing the hull of the ship; it is just mounted on the outside of the hull.
The Romans were extremely competent engineers and could have devised a system of gears, axles, pulleys, whatever to transfer the power generated by the wheel inside the boat to an underwater propeller that was outside the boat, and pushing the boat. (One simple idea: Turn the wheel 'sideways' so the axle is aligned with the long axis of the boat).
Multiple wheels could power multiple propellers; useful for steering or extra power when needed. Imagine 3 side by side: power the left wheel to turn right, power the right wheel to turn left, power left+right to go straight, power left+middle+right to go straight faster.
All the engineering skills and required materials for such a system were in place 2000 years ago; so the only missing pieces are inspiration and the power or money to turn inspiration into a working product.