Is there a location, in the southern hemisphere, where I could pitch a tent that I would be able to see the sun at any time of day for, say, more than a week?
How far north could I go and still be able to see the sun for two days or more?
The situation in the southern hemisphere is identical to the northern hemisphere, but they occur at different times of the year.
In the northern hemisphere the arctic circle is situated at 66.5 degrees North. Likewise in the southern hemisphere, the antarctic circle is situated at 66.5 degrees South. Between each circle and its respective pole 24 hours of daylight will occur for about a month or more from the onset of the summer solstice for each hemisphere.
The southernmost settlement in South America is Puerto Williams which is located at 56 degrees 56 minutes South. This is a lot further north than the antarctic circle.
To experience 24 hour days in the southern hemisphere you will have to be in Antarctica, as is evident from this picture, acquired from Wikipedia.
The Australian base at Casey is just outside of the antarctic circle having a latitude of 66 degrees 17 minutes South. However, the Chilean base at Carvajal is just inside the antarctic circle with a latitude of 67 degrees 45 minutes South.