If you are in ImageJ or QuPath you can use the Feret diameter, it is one of the Measurements available (not on by default) when you Analyze Particles.
And unless you have a Z stack, you could get the cross sectional area, but I doubt you would be able to get the surface area of the entire cell of a muscle cell. Just want to be clear we are talking about the same thing! Nuclei in the center are a bit easier, but the sum total of what you want to do is a bit more complex and I’d leave it to someone else to figure out how to link all of the steps together
To add to the fun, it looks like you might have the diaphragm set too tight/small on the light arm of your microscope, as I can see darkened areas at all four corners. If you are doing any kind of simple thresholding, dark areas tend to be problematic since they show up as all colors. The scale bar could also be problematic and should not be included in any image with automatic processing.
For the actual area, you would want to set the metadata in ImageJ using the Image->Properties. Once you have the pixel sizes, I think most areas should be generated in um^2 or some similar unit. The TIFF provided doesn’t seem to have pixel size metadata included. You might also be able to estimate it back in using the scale bar.
Most pixel classifiers (Weka might be an option if you are using ImageJ) should get you semi decent, though not perfect data (same errors as mentioned above, circled in blue here), with that kind of image. You can see why I would draw the area of interest to avoid the corners, though
*TLDR: The more automation you want, the better your original images tend to need to be. Or you need a lot of them (Deep learning). Human brains are still pretty good at providing context. Sometimes not as good as they believe, though…