First, assuming your curves and line segments aren't continuous, you'll want to weld them first. TIG-Weld which is available from the PluginStore (look at the top of this page) will do that for you. Then, when you select the edge in SketchUp, Entity Info will tell you the total length. You can type that into a label or other text box in LayOut.
Alternatively you could use SketchUp's Text tool to add a label that displays the length of the edge. Although that is simple, I'm not a fan of it because I prefer the appearance of labels made in LayOut and the options to control their appearance. LayOut's Label tool will display the length of an individual segment of the edge but not the total length of the edge.
Top label was created in LayOut. The bottom one in Sketchup.