Federal Reserve act of 1977. Dual mandate: price stability and maximum sustainable employment. Inflation around 2% and unemployment under 4% pre-COVID doesn't smell like failure to me, but that's up to you. Especially if you compare to other developed economies with even more sluggish inflation and insane unemployment rates since 2008.
That's not how employment works. There is a certain amount of natural unemployment even in a full employment economy: this is called "frictional unemployment." Like, you just moved or quit your job or got laid off and are looking for a new one. You are unemployed for a couple weeks or months, or longer if you're retraining. There's also structural unemployment, people who are unemployed (or underemployed) because their skills are not suited to the current environment. Central banks can't do anything about that with interest rates. There will always be some people unemployed.
Full employment is defined by the concept of the NAIRU, the non-accelerating inflation rate of unemployment. The lowest sustainable unemployment before inflation begins to take off in a feedback loop (as higher inflation pushes expectations of future inflation up, resulting in a spiral).
Yeah, there's a big fight about this - AngrySkillz is entirely correct about "NAIRU", but it's one of those parameters that's difficult to measure because you can only find out when it's "too late".
Heck, even the "4%" is a percentage of people "looking for work", which is obviously a quantity that can vary depending on how good or bad the outlook is.
