If you suspended your DNA like a rope in its 6ft full length, could you feel it if you waved your hand through it?
Research on tactile detection has found humans can detect wrinkles in a surface on a nanoscale. One could argue that we may be able to feel a strand of DNA. 

However, this research found only wrinkles of 10 nm were detectable. Therefore it is unlikely we would detect a DNA strand by simply touching it, considering DNA has a radius of one nanometer radius (10 ångströms).

We still may feel a pull of the strand depending on the tensile strength and bond strength but I'm not sure how those values relate to human perception.