If you are in the blast zone for any nuclear weapon you cannot modify your usrrent structure to withstand the blast. Well ...I guess you could ...maybe ...'cuz anything is possible with enough money ...but realistically ...no.
But ..if you want a "secure" shielded" space that will provide sufficient shielding from ionizing radiation ...then yes ...you coul do it. And lead is indeed your preferred choice. Its NOT the cheapest or lightest ...but it is the thinnest. It takes more water, concrete or poly to provide the same shielding as a few inches of leach can provide. Which makes it easier to cover up with the design of the house. However ...lead is heavy... and thus needs support both vertically AND horizontally. But you could (at least in a lot of slab-on-grade homes) cut the floor ....backfit footings to carry the weight and create a steel framework to hold the lead. Commercially, you can buy lead in sheets and bricks. I don't know where you'd get it, but they do make it.
Underground , though, would be a better and more cost effective location in the long run...since its easier to design for blast resistance but just as important as there is significant concussion form a blast that crush below grade structures. That is why many of them are shaped like a sphere or an egg since the design for naturally is stronger.
I like the driveway idea mentioned above ... 'cept its hard to hide what you are doing from your neighbors. But is close ...not much in the way of utility services typically runs under it and you could possibly retrofit an interior egress point that could connect to it by underground tunnel. Though all good rabbit holes need at least two entrance/exit paths. As for cars driving or parking on top ...that can be resolved with a properly engineered slab that when finished would look like everyone elses driveway.
There are no cheap alternatives ... well ...without sacrificing something.
Personally ...I consider nuclear attack farther down the lsit than other potential disaster scenarios ....and strive to address being prepared for them and then ..once done ...would try to confront that possibility. And I would chose underground. There are several frefab systems available and given that I would covet the secrecy factor I'd install min before my house was built and would chose a more remote location so as to have fewer people that could see what I'm doing .... that way I can chose who knows my preparatins and who doesn't.
Good Luck !