To measure latitude, you must measure the elevation of some celestial body. Basically, you will use the the Sun, or stars (trajectory of planets and the Moon is too complex to serve much here).
If you use the Sun, then you are using a projected shadow (you don't look at the Sun directly). You have some big pole, that you try to erect as vertical as possible; and you measure the length of the shadow at noon on the equinox. You will need to do some year-long measures to work out when the equinox actually is. The crucial point is that the Sun is not a point in the sky; it has an apparent diameter of about 30' (half a degree). This is the reason why, when you look at the shadow of some building, projected on the floor, the shadow edge is blurry: this transition zone between shadow and non-shadow corresponds to the floor spots from which the Sun is partially visible, and partially hidden by the building. The bottom-line is that a Sun-based measure of latitude tends to be imprecise: accuracy is within half a degree, but no better. (With a sextant you can have a much better accuracy, but that's because that apparatus includes filters allowing the operator to actually look at the Sun, and target the disc edge, instead of "the Sun in general" as in a shadow-based measure.)
With stars you can do potentially better, because they are points (at least to the naked eye) and you can stare at them directly without going blind. If you use stars, then you must follow several for a night, noting their azimuth and elevation throughout the night: this is sufficient to recompute their apparent trajectory, and then work out the latitude. The accuracy of the human eye is, at best, 1' (1/60th of a degree). However, it is hard to achieve in practice.
Notably, even if you can see a 1' angle deviation, the measure will depend on the accuracy with which you know the geometric characteristics of the device you are using (including the measure of "vertical" and "horizontal"). Also, prior to Gauss and Legendre in the early 19th century, astronomers had no systematic method to deal with measure errors and smooth them out with averages and statistics.
As a data point, Tycho Brahe, in the late 16th century, achieved measures with about 2' accuracy on average. These measures would translate, indeed, into a computation of latitude with the same accuracy. It must be noted that Brahe had very good eyesight, was exceptionally stubborn, and benefited from the precision offered by late Renaissance instruments when it came to measuring the length of, say, a ruler (according to David S. Landes, we have to thank clockwork technology for the availability of such tools in the Renaissance).
As another data point, the Great Pyramid of Giza (built circa 2560 BC) is aligned on the cardinal point within 4'.
From all these informations, we may conclude that astronomers from around 1 AD could achieve a measure of latitude with an accuracy of about 4' or so, but at considerable expense. Hipparchus has apparently done so in some occasions, but he had dedicated his life to such matters.
Longitude is much harder: it can be measured by the difference between the local time and a reference time. If the Sun appears to reach noon while your watch says 2 o'clock (while it matched the Sun in your home town), then you know that you went 30 degrees to the West. This is about the only direct way to measure longitude: you need to bring a clock with you, and you will get as much accuracy as your clock provides, with 1 degree of longitude for every 4 minutes of time. Since clocks in antiquity were awfully inaccurate, this was not workable at that time. Indeed, longitude is measured by the difference in time between a sundial (which measures local time) and a clock (set to the reference time). When the clock is less accurate than a sundial, it is quite hard to reach any conclusion at all.
Some indirect measures can be done in rare occasions but need astronomical devices that were not available at that time (e.g. telescopes to observe the transit of Venus when it passes between the Sun and the Earth).
All longitude estimates in antiquity use the indirect method by which the longitude is inferred from the actual land distance, obtained through some other mean (mostly triangulation with notable geographical features such as hills and buildings). This works well with short distances (e.g. between Athens and Corinth), much less for long distances, and very badly when a sea is involved. Contrary to latitude, astronomers in antiquity could not get a notion of absolute longitude, only relative for locations which are close enough to each other.