My guess at a possible approach would be to launch a cube sat using some sort of grant. Use high res thermal imaging from the satellite to detect missiles near their apex (unless you could somehow use starlink backscatter). Then use a series of ballons launched in such a way to maintain coverage near your area carrying high energy lasers (or maybe a maser) and their own thermal imaging to help calulate/triangulate for targeting.
Who know if that would work. Im completely makjng this up. That would probably be the cheapest way to do that since the image processing would be more accessible than quality radar and associated processing. You also wouldn't need extremely expensive missiles. The most difficult part would be ballon recovery to reuse the expensive equipment.
You know, things are tricky in recent times
You need a powerful radar, at least this big
https://www.radartutorial.eu/19.kartei/06.missile/karte013.e...
and also at least one solid-fuel interceptor missile which can reach a height of 10-200 miles depending on how far away you want to hit it. There is a close coupling between the radar and the missile in any practical ABM system.
The current approach to missile interception is to get a direct hit, the US has had an exoatmospheric kill vehicle since the 1980s that has been used on antisatellite and antimissile missiles.
Exoatmospheric intercept may be tricky though because it's believed the warhead will be accompanied by a large number of decoys which could be as simple as mylar baloons. The warhead is heavier than the decoys so when it reenters the atmosphere it can be separated from the rest and destroyed... with very little lead time. The 1950-1970s doctrine was to shoot down a missile with a nuclear warhead. That's kinda the worst case to generate a nuclear EMP so the ABM system could very well blind or daze itself with the first strike.
Today there isn't a lot of talk about the decoy problem, either they think they have sensors good enough to find the real warhead or they are deliberately ignoring it.
One challenge is "boost-glide" warheads that can slide 50 miles or more horizontally when they reenter like the old Pershing II or China's DF-21. It is relatively straightforward to add "boost-glide" to a missile with a sub-1000 mile range. A boost-glide ICBM was tested by the US circa 1980 but it is much more challenging. This is a problem in many respects, one of them is you can't tell what the missile is targeting until it gets very close.
And yeah, you need some place where you can test it so you'll need to go to some desert island and intercept missiles.
I do not have the answer you are looking for but I would suggest a much less expensive alternative would be to use the earth under your garage as the interceptor and build a hardened bunker deep under ground. It's still going to be expensive to do it right and it will need to facilitate healthy living under ground for a long time but does not require the permits one would never get to build missile interceptors.