I have been sick for two days.
In practice, this means that you got sick sometime the day before yesterday and are still sick. With the perfect tense the state is still ongoing. However, English speakers are not that specific about the start point of the state, though, so it is possible you are referring to some other day than the day before yesterday.
For most units of time, a good guideline to follow is that "two X ago" can mean anywhere between "one and a half X ago" and "two and a half X ago". For instance, if I said, "I have been living in city x for a year," I could mean ten months, or I could mean sixteen months. If you want to be more specific, you use a more specific unit of time, such as, "I have been sick since Thursday afternoon" or "I moved to this city eleven months ago / in January last year."