First, let's get the scale bar out of the image. There are a few ways to do this (`MorphologicalBinarize` and friends), but I went with the more eye-bally approach of using `ColorReplace`. Here, we're replacing black pixels with white, and all other pixels with black, to get a mask. Note that ColorReplace has a third argument `d`, which you could use to fine-tune what it sees as "black".

    i = Import["https://i.sstatic.net/yR6UY.gif"]

    c = ColorReplace[i, {Black -> White, _ -> Black}]
    
    
[![enter image description here][1]][1]

Compare with a simple `MorphologicalBinarize` - the choice is yours:

[![enter image description here][2]][2]

Now we can get the bounding box of our rectangle, and simply take the difference:

    box = ComponentMeasurements[c, "BoundingBox"][[1, 2]]
    
    Differences /@ Transpose@box

> `{{38.}, {5.}}`

So the longest dimension of our bounding box is 38 pixels long, and the short side is 5 pixels long.

Just to check, I counted the pixels another way, using `SequenceCases` on our replaced image, and each line of `1`s, the pixels of the scale bar, had a length of 38.

    SequenceCases[
     Flatten@MorphologicalComponents[c], {p : Repeated[1]} :> Length@{p}]

> `{38, 38, 38, 38, 38}`

Either way, we get about 38 pixels, so now we know the length of the bar, and thus we can get the size of a pixel (assuming pixels are square here).

    pixelLength = Quantity[3, "Microns"] / 38

> `Quantity[0.0789474, "Microns"]`

and now if we get the Euclidean distance between your points and multiply it by our pixel length, we get the distance in microns.

    EuclideanDistance[{5.5, 39.5}, {13.9828, 37.5345}] * pixelLength

> `Quantity[0.687437, "Microns"]`

This answer will not hold up well if the bar is not always aligned perfectly to the image border, as it is in your test image. You could consider `ComponentMeasurements[c, "Length"]` or `"CaliperLength"` to get the length of the bar instead in that case. However, it may take the distances between opposite corners of the bar, which may not be what you want.


  [1]: https://i.sstatic.net/EQjfL.png
  [2]: https://i.sstatic.net/2z9bv.png