Here's one way to do it. Though I don't know if it is the "proper" way:
-First convert your stream order layer to a polyline from: Arc Toolbox > Conversion Tools > From Raster > Raster to Polyline. Add your stream order layer as an input and uncheck the "simplify polylines" box.
-Next add a new field to the attribute table of your newly created layer: Arc Toolbox > Data Management Tools > Fields > Add Field Choose your newly created layer as the input table and enter "length" (or whatever) as the field name. choose "LONG" as the field type.
-Now open the attribute table for the layer created in the first step, right-click on the field you've created and choose "calculate geometry". Choose your units and click OK.