Quote from piezoe:
It's a good question. It would require a very small reactor, however it seems several factors, in addition to the safety concerns, the power to weight ratio, and the public fear of anything with the name nuclear attached, no matter how irrational, make it impractical with today's technology, .
Probably somewhat more likely than a car powered by a conventional water cooled reactor would be a small thermonuclear heat source in conjunction with a solid state device like a thermopile to produce electrical power which would constantly charge a lithium ion battery.
In the 1970's research was carried out aimed at producing a Pu-238 powered heart pacer, based on this same principle of using heat to produce electricity from a solid-state device. The idea was to make a pacer that could be implanted and would outlive the patient. As pure Pu-238 is an alpha emitter, there is no danger of radiation from Pu-238 itself, however over time daughter products from Pu-238 disintegration would grow in and the radiation would slowly become a mixture of high energy photons (gamma), electrons (beta), and some neutrons as well as the harmless alphas. So the question would be how harmful is this radiation over time compared to the benefits of a very reliable, long lasting pacer. Would the patient be dead from natural causes long before this radiation posed any significant risk?
Far more likely than nuclear powered cars at this point would be more nuclear powered trains and rapid transit, though even their it would seem most practical to simply go from diesel to electric power and generate the power at a centrally located nuclear power station, as is done with the current technology. The major disadvantage of centrally located power stations is, of course, the large power loss attendant on transmission of power over long distances due to the impedance of miles of high tension lines. Although these losses are reduced by transmitting at high voltage and then stepping down to the end user, the losses are by no means insignificant.