Thank you! I don't have exact figures handy but short answer no, not at this scale at least. Manufacturing, design, materials, safety etc. all easily runs in the tens of thousands of dollars. We are on a much tighter budget, but AFAIK USC has spent hundreds of thousands of dollars on a single launch.
It has a lot to do with how much of a craftsman you are too. The really good people in the high power rocketry hobby can get above 100k feet on a budget of maybe $3-4k
These rockets are almost always 2 stage designs. The most expensive parts are the motors and electronics. Typically, you'll have redundant flight controllers for the booster and sustainer plus GPS/radio trackers to find them once the rocket eventually lands.
They aren't much different than the typical little hobby shop rockets just bigger and faster. The ones i've seen were maybe 8-10' tall, hand rolled carbon or fiberglass fuselages, and peak at around Mach 3.
Yes, in theory, if you happen to be an extremely talented rocket engine designer and architect and have about ~50-100 K lying around, you could in theory build a rocket that could cross the karman line. With modern technology, you could probably clone a V-2 if you were born a genius.