Scientists at NIST took 48 measurements of the atomic weight of a reference sample of silver using two nearly identical mass spectrometers. This project was undertaken in conjunction with the redetermination of the Faraday constant.

Reference: Powell, L.J., Murphy, T.J. and Gramlich, J.W. (1982). "The Absolute Isotopic Abundance & Atomic Weight of a Reference Sample of Silver". NBS Journal of Research, 87, pp. 9-19.
Source: http://www.nist.gov/itl/div898/strd/anova/Ag_Atomic_Wt.html

This dataset contains 29 measurements of the density of the earth, obtained by Henry Cavendish in 1798 using a torsion balance. Density is presented as a multiple of the density of water.

Reference: Moore, David S., and George P. McCabe (1989). Introduction to the Practice of Statistics. Original source: Stigler, S.M., "Do robust estimators work with real data?" Annals of Statistics, 5 (1977), pp. 1055-1078.
Source: http://lib.stat.cmu.edu/DASL/Datafiles/Cavendish.html

Duration in minutes of the eruptions of the Old Faithful geyser in Yellowstone National Park.
108 measurements
Source: http://www.stat.sc.edu/~west/javahtml/Histogram.html

The first 5000 digits of the mathemtatical constant pi (= 3.1415926535897932384...) were reported in Mathematics of Computation, January 1962, page 76.

Interesting questions involving pi digits:
1) Are the digits uniformly distributed?
2) Is there serial correlation between successive digits?
Source:
NIST National Institute of Standards and Technology
Statistical Reference Datasets
http://www.nist.gov/itl/div898/strd/univ/pidigits.html