Answer
Less than 8 km/s.
Work Step by Step
This is similar to a Newton's cannon problem. 8 km/s is the required speed for a Newton's cannon on Earth (page 190). Eros has a weaker gravitational pull than Earth, which decreases the necessary speed, because the cannonball would drop a smaller distance in the same time.
Moreover, because Eros is much smaller, the ground is more severely curved than Earth is, so the cannonball could be fired much slower and still "fall around Eros".
A quantitative analysis shows that the required speed of Newton's cannon is the escape speed of the planet divided by $\sqrt{2}$, so the trends in Table 10.1 also confirm the answer.
