Red Thunder by John Varley
As far as SF is concerned, the story is about going to Mars. As has happened before, the US has been sitting on its laurels while China builds a space program which includes the foundations to send a mission to Mars. Once the Chinese actually launch a slow ship to Mars, the US does a rush job to send its own ship to Mars. Even with a faster ship, the Americans won't get there first, but they may be able to get there almost as soon and get back to Earth first. That will at least minimize the public relations losses.
However, in the rush to send a ship, the US has decided to use a new propulsion system which may fail when used in the way needed to get there quickly. The seriousness of this issue is discovered by a relative of one of the astronauts. Between a concern for the lives of the astronauts and a wish that Americans could reach Mars first, a small group of people in Florida start to plan their own private mission. They plan to use a recent physics discovery one of them has made in order to use an entirely new way to propel a rocket which would allow them to reach Mars in less than a week.
Some readers may find the above scenario unlikely. Keep in mind I phrased it to make it sound more plausible than it actually is in the book. (See the end of the review for a less generous description.) The story is written such that the far-fetched aspects are no more offensive than is unavoidable, but be prepared to give your suspension of disbelief some extra elbow room to work in.
Some elements of the narrative style reminded me of the perspective, imagery and patter of a Sam Spade kind of detective story, in other ways it's more like the lower-middle income White Floridian odd-ball fiction that some authors have been writing. The style works with my tastes. However, it's not a typical SF style and may not work with some SF readers. Whether or not the narrative fits your tastes may be important - the book doesn't start dealing seriously with the various Mars missions or the planning phases for building the private rocket until you're one third of the way through it.
This is not what I'd call hard SF. I find it difficult enough to believe someone could take a common household electronic device and jerry-rig something that would create phenomena involving extra dimensions. But you don't have to be a rocket scientist to figure such a device is not going to fiddle with extra dimensions or compress materials under extreme pressures if you don't even have two 1.5 volt batteries to power its functioning. Nevertheless, that is a central premise to the story. Perhaps, that is the most blatant example, but it's by no means the only implausible element.
I could recommend this book to someone looking for light entertainment, but not for someone in the mood for serious SF.
The less generous description:
The story has a group of around ten people spending somewhat more than $1 million and less than 6 months to plan and build a space-worthy rocket ship. The budget is able to be so small partly because of their implausible propulsion system. But considering what $1 million will buy in terms of buildings on Earth, how can it build a spaceship? They do have an ex-astronaut to provide a skilled space pilot. They have a genius for the propulsion system. They have a skilled electrician/electrical engineer, a skilled welder, someone with experience repairing and rebuilding road vehicles and someone skilled at running business operations. This is about the extent of their strengths in knowing how to put together a space expedition from scratch. Nobody in the group has experience designing spacecraft or even designing ocean-going ships or large buildings. Some of the central individuals are college-age kids who don't yet have a college degree. The project to build the rocket is being done in secret, which places some restrictions on their efforts.