measuring distance

There are a number of websites that let you draw a track on top of OpenStreetMap based maps (as well as other maps), then tell you the distance. Some of these are listed here: Track drawing websites

Its also possible to do this with software on your own PC, using OSM based maps. One option for this would be using QLandKarte GT. Or there's a variety of a other software that could work, depending on your operating system and exactly what you want to do.

Your question is not OpenStreetMap specific; the distance between points is independent of the underlying map, and would usually be caclulated in the application that displays the map. There is a cycle route planner that uses OSM map tiles and lets you plot a route - but if you use that application, then you do not "measure distance in OpenStreetMap", you use OpenLayers/Javascript to measure distance between locations, and you could just as well display any other map below!

If you want to measure distance in an OpenStreetMap editor, use JOSM which has a measurement bar beneath the main window, and there's also a measurement plug-in for more refined tasks.

Theirs a company called mapyx/quo that as openstreetmap as an alternative to the ones they sell. you can use it for free. It as the usual mapping tools you would need to plan routes I'm pretty sure it will import and export to some gps units.I know its not exactly what you asked but may be of some use

Find more api details with exact script here.