Beaufort West is the archetypical Karoo town. Surrounded by farmland and approached on a long, straight road from either direction, it's almost an oasis. Many people just zip through here, stopping only for fuel and a snack, but it's an attractive town with an interesting museum complex.
Most of the exhibits relate to the life, history and career of one of the town's most illustrious sons - the heart transplant pioneer surgeon, Chris Barnard. It is also a very convenient overnight stop if driving between Cape Town and Johannesburg. It's about 500km from Cape Town and 900km from Johannesburg and there are a number of inexpensive accommodation options.
A far more interesting place to stay, though, would be the wonderfully scenic Karoo National Park, which is worth a visit for its own sake. It is conveniently situated about 10km on the Cape Town side of the town. With lots of different birds, many species of Antelope, and Black Rhino, it's an attractive and exciting place to stay.
Even if you only plan to overnight, it is worth fitting in a quick game drive or a walk. A bonus is the short interpretive 'hands on' fossil trail, which is wheelchair accessible and has interpretive Braille plaques.