An comprehensive source on this issue is Brickers' Astronomy in the Maya Codices. They meticulously correlated Mayan calendar with the Western one, and systematically analyzed their Venus, Mars and eclipse tables. It turned out that a table dated to 11-12th century predicted a solar eclipse in 1991 within a day of actual occurence. Another measure of accuracy is given by comparing Mayan values to Ptolemy's and the modern ones, and one can see that they did essentially as well as Ptolemy:
Parameter (days) | Maya | Ptolemy | Modern
Lunar (synodic) month | 29.53086 | 29.53337 | 29.53059
Synodic period of Venus | 583.92027 | 583.94267 | 583.93
Synodic period of Mars | 780 | 779.94 | 779.94
Solar (tropical) year | 365.242 | 365.24667 | 365.24198
Mayan observations were performed with a naked eye assisted by a pair of crossed sticks, and they did not have any kind of angular coordinates to record and predict positional data such as found in ephemerides. All the tabulated data in the surviving codices is on cycles, time periods between recurring astronomical events, solstices and equinoxes, zenial passages, eclipses, risings and settings, stations and so on, and it is about this data that the accuracy claims are made. For instance, the prediction error on eclipses and Venus's revolutions is under one day in 6000 years. Apparently, they even noticed a regularity in the motion of Mars that went unnoticed until recently found in the codices. Some numerical data from the codices is reviewed here.