I agree with you that a short circuit is what I expect for DC resistnce. TV antennas can have a folded dipole as the driven element which gets the impedance into a better range, or a split dipole or several different antennas mixed on the same pole. This link shows both types.
http://www.yellowpages.com.au/articles/television-antenna-services-27308/how-high-should-i-mount-my-tv-antenna-20121017t120043n00f0032
The basic configuration of the split type is open circuit, and the folded type (with the elongated loop) is a short circuit. These will very likely be connected through a broadband transformer (balun) and then the coaxial cable down to the wall socket, which is an unbalanced cable. A short circuit would be a few ohms from the room end, and is what I would expect to see. The usual fault is corrosion around the terminals at the antenna or balun transformer connection so it could well be open circuit. Try when wet on the highest ohms scale? If it changes it sort of shows the cable is ok.
If there is a masthead amplifier it needs to be powered from the bottom somehow. Here is a balun transformer, connected right at the antenna to match it to 75 ohm coaxial cable..
http://bourneelectronics.com/Universal-TV-Antenna-Balun-P1483329.aspx
If you have a capacitance meter (on the multimeter) you can get some idea of the distance to the open circuit if you know the cable type. e.g. 75 ohm polyethylene foam TV cable (e.g. RG6) is something like 50pF/meter or 16pF per foot. This can be confused by a break in the outer sheath or wire though. The other idea is look with binoculars.
You could always call a TV antenna person. With changeover to digital the antenna itself might need attention.