I assume you mean it's impractical in terms of all the changes that would be required to get from the current U.S. system of government to a jury duty republic system because of the significant changes to the Constitution that would be required and the near impossibility of passing constitutional amendments, particularly given the extreme opposition that the entrenched ruling class would have towards such a system (because the primary goal of such a system is to uproot an entrenched ruling class and replace it with something more egalitarian).
However, suppose for a moment that you're the revolutionary leader of a small nation in Latin America and, after a long struggle, you have risen victorious over the oppressive old regime of your country and there is overwhelming public support for you to become dictator. But, being aware of the inherent long term problems of dictatorships, you don't want to be a dictator. So, you declare, "We will implement a constitutional republic like the Americans. But, to ensure our government always has the best interests of our people at heart and we are never again dominated by an elite ruling class, we will choose our representatives in the legislature by a lottery like the American PowerBall." What, then, would be impractical about implementing a jury duty system of democratic representation?
Problems that I see revolve around the formalities of office. Learning your way around legislation, the formats, styles, trade-offs, and compromises that must be made all require time.
Beyond that, you run into the problem of lobbyists. Not the evil type that is a stand-in for the word "corruption", but instead the legitimate "education" type lobbyists. You lose much of the institutional knowledge that prevents candy coated views of the world from taking hold. Honestly, how many of the randomly selected people are going to understand the trade-offs of each and every policy, be it economic, societal, political, industrial, trade, environmental, etc. Having experienced people who have been around a few cycles provides a base level of knowledge.
However, suppose for a moment that you're the revolutionary leader of a small nation in Latin America and, after a long struggle, you have risen victorious over the oppressive old regime of your country and there is overwhelming public support for you to become dictator. But, being aware of the inherent long term problems of dictatorships, you don't want to be a dictator. So, you declare, "We will implement a constitutional republic like the Americans. But, to ensure our government always has the best interests of our people at heart and we are never again dominated by an elite ruling class, we will choose our representatives in the legislature by a lottery like the American PowerBall." What, then, would be impractical about implementing a jury duty system of democratic representation?