First, I'll point out that the article Pete refers to is also available on the USATF site. You can start from www.rrtc.net and choose the Tools for Course Measurers link, then scroll down to More Publications from USATF Road Running Technical Council, then choose How to Tape a Track.
I was the author of the indicated article, but the technique was originally due to Bob Letson. In particular, while the basic idea of measuring a length and width and assuming ideal track geometry is fairly obvious, it was Letson who developed the technique of measuring two widths (W1 and W2) and averaging them. I think this improves the accuracy considerably.
I did a comparison of direct circumference measurement of a track and length-width measurement, described in Measurement News #17 (the June 1986 issue of MN) which is available in the Course Measurement archive Pete referenced. See the article "Measurement of Memorial High School Track, Tulsa, OK" on pages 12-20, describing measurements I did with the late Glen Lafarlette in March 1986. On that occasion, the discrepancy between length-width method and direct curb taping (when using the same tape) was only 8 millimeters. Some points to note:
As explained in my description of the length-width method, the two widths (W1 and W2) should be chosen near the ends of the straightaways, but within the straightaways, as you shouldn't waste time trying to locate exact junctions of straightaway and curve. In that 1986 measurement, we measured along 5-yard lines of the football field. Having a football grid marked in the infield is extremely useful to make sure you're measuring widths perpendicular to the straightaways. Without such a grid to guide you, you'd very likely overestimate the W1 and W2 distances, thereby overestimating the track length.
For marking intermediate tape lengths when measuring across the grass infield, we used a handy technique developed by Wayne Nicoll.
We attempted to locate one of the radius centers in order to check the geometry (to see if the radius was constant as Mark suggested above, and as Pete suggested many years ago). Locating the radius center was far more difficult than you might think. We kept encountering big discrepancies, which we realized were happening because we hadn't located the center correctly. After a number of attempts, we decided it was hopeless, and gave up.
Mark indicated above that the length-width method would underestimate the track length if W1 differs from W2. I think this is correct, but I estimate the amount of underestimate as about:
(W1-W2)^2 / (4*S)
where S is the estimated length of the straightaways, i.e.,
S = L - W
where W is the average of W1 and W2. In that measurement I did with Glen in 1986, the difference between W1 and W2 was about 2.7 cm, while the track straightaway length was about 96 m, so this formula suggests the amount of underestimate as about 0.002 mm (or 2 micrometers). If the W1-W2 difference were as high as 20 cm, the amount of underestimate would still be only about 0.1 mm.