Typically, you see negative impedances when you are looking into a source. ADS shows negative values when you put a Term object on an oscillator, and look at it's output impedance. If you look at how S11 is defined, you'll notice that the signs are opposite, since you are supposed to be driving current into the object, and not getting more current out (which is what an oscillator does, produces power).
My guess is that your antenna is picking up a signal, so your network analyzer is seeing the same thing ADS would... put in 1 mW, and you get out 10 (makes the answer look goofy). Try shielding your antenna from external RF sources. Depending on your frequency range, that could be difficult, since the wavelengths may be long. Point it away from fluorescent light fixtures, which are noisy and can have a wide spectrum of noise. See if anyone else in your lab, or nearby, is transmitting on/near your frequency range.
Depending on your antenna design, you should have an idea where your first resonance should be... you should have designed around that point/value. It may be somewhat off in reality, compared to your design equations, but it should be close.