In the spirit of counterfactual rocket design questions such as this one - how would the STS have differed from its actual design if it used J-2's as the main engine? What would be the loss in payload mass assuming a straight replacement, and what kind of design would be necessary to achieve the same performance as the actual STS?
The J-2 was not suited to firing at sea level. A sea level specific impulse of 200 seconds is widely quoted for the engine, but I'm suspicious of such a round figure; the exit pressure of the engine might be too low to fire at sea level without flow separation.
Assuming that it could be fired at sea level, or air-started, a scaled-down version of the orbiter might have been feasible at 10-12 tons of payload to LEO as compared to STS's 27.5 tons. It would need 6 J-2s instead of 3 SSMEs, which would make for a rather bulky tail end, and would still have the full-sized boosters and external propellant tank.
It was at one time considered to make a version of the J-2 using a toroidal aerospike nozzle instead of the bell nozzle, which would have much improved low-altitude performance. Again, scaling down the orbiter, 20 tons or so of payload would be feasible, but again you'd be faced with the problem of finding tail area to mount 6 engines. I think the J-2T concept also lacked thrust vectoring (gimbaling) capability, so maybe some compromise like 4 aerospikes and 2 standard J-2s would be necessary.
As discussed here, some extensible-nozzle J-2 variants were also proposed, simpler than the aerospike, but they are even larger at the nozzle exit plane, so, again, the orbiter configuration would have to be much different from the STS orbiter we're used to to fit them.
$\begingroup$ For those wondering what the aerospike nozzle mentioned looks like. $\endgroup$ Jul 9, 2019 at 13:15