ἀσθένεια has a semantic range that includes physical weaknesses, such as those you noted in the title (weaknesses, diseases, sickness), and also soul weaknesses. See G4561.
The use of the word σάρξ (flesh) in this verse indicates that ἀσθένεια is being used in a physical sense. Some scholars believe that this was an eye problem (See this commentary), although opinion is not universal. An interpreter would not be in error to state that Paul was certainly suffering from some kind of physical weakness, and can infer that this would be a sickness of some kind that was noticeable to the Galatians, since they consciously received him in spite of it (Gal. 4:14).
In summary, the text does indicate that this is a severe physical sickness, which very well may be an eye problem.