Zero unemployment is theoretically possible, but economists would say that it is a bad idea.
It would be possible to ensure that everyone had a job. All that would have to happen is that the government would have to guarantee that it would hire anyone who could not find a job. The government could simply create “make-work” jobs that had little economic value. They could have lots of people sweeping the streets, for example. Or they could hire large numbers of janitors to keep schools and public buildings clean. With everyone who wanted a job employed, there would be no unemployment.
Economists would say, however, that this is a bad idea. This would be true for at least two reasons. First, with no unemployment, wages would be pushed up and up. Companies would have to compete very hard to hire people because there would be no pool of unemployed people. Second, and more importantly, there would be little turnover in terms of employment. Companies would be very reluctant to fire bad workers because they would not be sure that they could find good workers to replace them. This would lead to a less efficient economy as people would not naturally gravitate to jobs they were good at.