Standardization is a tricky process even in (perhaps especially in?) the age of electricity and computers. Essentially you need some sort of agreement that means people are going to coordinate their basic measurements. Sometimes this comes from the "top down" (e.g. the state), sometimes it is an accord across an industry (many "industrial standards" are just protocols agreed upon by participants in an industry without government interaction).

Anyway, in principle standard weights and measured are "guaranteed" by the production of prototypes to measure against. So the fact that in a vault somewhere is a "prototype meter bar" and "prototype kilogram" is pretty well known, though those are late-18th century versions of activities that have gone on for a long time. Basically you take a length of material and say, "this is now an X" and then you make copies of it and distribute them to the various people who might use such lengths as the basis for their own measurements. This approach is very old — the ancient Egyptians had "cubit rods" which served this purpose, analogous to how one might use a "meter stick" or a "yard stick" today. Later standards are often derivable — e.g., you can make a fairly standard mercury thermometer based on the definitions of the celsius or fahrenheit scales alone (water freezes at one point, boils at another, and mercury expands pretty uniformly between these points), without reference to an original. (Base SI units are now defined in this way, or derivable from other base units — e.g. the definition of a kilogram was derived from a definition of a meter which was itself based on a fixed percentage of the Earth's radius. The tricky thing in basing units after measurements of this sort is that measurement quality can itself change over time, as it did in the case of the meter.)

But really one must also say very clearly that they weren't at all "guaranteed," well into the modern period. Material fabrication has its tolerance limits, and that assumes you are actually making the right copy in the first place. Materials can warp with age and use. By modern standards (getting a little meta here?) many of these old "standards" weights and measures were off by a considerable amount (the original definition of the "meter" was based on faulty measurements of the Earth, as well). The difficulties of this lack of standardization was the start of many of the first "scientific" agencies in governments in the late-19th and early-20th centuries, like the National Bureau of Standards (USA, 1901), because there are real consequences for lack of standardization (in can result in major economic problems, as well as major health and safety problems). Standards-sensitive industries (like railroads, who require very precise time standards to avoid accidents) would often develop their own agreements with or without government participation (standardized time zones, for example, were originally an innovation of the rail industry). But standards screw-ups are still with us today — consider that the Mars Climate Orbiter burned up in the Martian atmosphere in 1999 in part because some engineers were using Imperial units while others were using metric!

Further reading: a great discussion of the problems of standardization, and putting them into a historical context, is Ken Alder's The Measure of All Things, which revolves around the rather fascinating story about how the measurement for the standard meter came about just after the French Revolution. I might also plug the book of one my colleagues, Andrew Russell, Open Standards and the Digital Age: History, Ideology, and Networks, which talks about standardization processes in general, but also with respects to more modern developments (like TCP/IP), which are no more "simple" than the historical processes despite the apparent advantages that better communication technology might bring (indeed, the standards that made the Internet have a lot of counter-intuitive aspects to them). Lastly, Peter Galison's Einstein's Clocks, Poincaré's Maps, puts issues with time standardization at the heart of some of the major developments in early 20th century physics — an interesting and perhaps unexpected intersection. Separatly, Hasok Chang's work on the history of temperature (Inventing Temperature) gets into some of the very thorny practical and epistemological issues that show up when you start trying to make standards about phenomena you don't completely understand to begin with (like temperature — people were making thermometers well prior to the modern understanding of heat, and that produces a lot of questions like, "what's an ideal fluid for a thermometer, and how would we know? If we're not sure what heat is, how can we know we are measuring it correctly?").