It is currently possible to achieve sub-centimeter accuracy when surveying a point.
To express a latitude and a longitude, it needs to be reference to something. We need a datum: a reference frame and a reference ellipsoid. The reference ellipsoid is simply the mathematical shape that we use to perform the conversion to a geographic coordinate system (lat/long/height). The reference frame is the set of physical points (stations) that serve to locate and orient a coordinate system. For example, the ITRF2014 (International Terrestrial Reference Frame) is defined with the coordinates of hundreds of stations around the world, as well as their velocities (because of crustal movements). The ITRF2014 is globally non-rotating relative to the crust, is centered on the geocenter, and has no translation. It is an ECEF (Earth-centered, Earth-fixed) frame of reference.
This most recent realization of the ITRS (ITRF2014), with modern geodetic techniques (mainly VLBI, DORIS, SLR and GNSS) has a very good overall accuracy and consistency, and the errors are typically only a few millimeters. Countries often use other local or national datums to express positions, using reference frames that locally follow the crust and reference ellipsoids that better approximate the shape of the Earth in that country. Accuracy and consistency of these datums can vary a bit depending on which techniques were used to realize their reference frames.
Now for the practical measurement part, if you wish to determine the lat/long of a point, RTK GPS, and even traditional surveying techniques can be used to achieve a reasonable accuracy of a few centimeters pretty quickly. These techniques require a nearby known reference point or broadcasting station for GPS, usually from a densification network, a local geodetic network or some other previously measured point. With more advanced GPS survey, if you remain static for several hours or days and collect enough data from satellites and post-process your data using nearby ground station corrections, it is generally possible to get down to a few millimeters in accuracy. Sub-centimeter accuracy surveys can help analyzing crustal deformation, tectonic motion, isostatic rebound, sea level rise, etc.