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Yes, OmniGraffle has support for doing things like this.

First thing you want to do is to set up a drawing scale and units for your canvas, with the Canvas:Size inspector (cmd-3). You may also want to turn on the grid display (same inspector, Grid tab), and the smart guides (Arrange->Guides->Smart {Alignment|Distance} Guides).

Make yourself a pair of horizontal and vertical lines with arrowheads that look like |< and >| by drawing lines, selecting them, opening the Style:Lines and Shapes inspector (cmd-1) and choosing the arrowheads. If you hold down the shift key while drawing them, you'll get perfectly horizontal or vertical lines. Next, select one of the lines, option-double-click on it, and it should create a line label for you. Without clicking anywhere else, use Edit->Insert Variable->Line Length to make that into a magic line length variable which displays the length of that line. Repeat with the other line. Now you can just duplicate those lines and drag them to any place where you want to show a measurement on your diagram, adjusting them to match the size of the object or distance being illustrated. Be sure to drag on the line itself or you'll just pull the label off...