Radio Astronomy and SETI
Believe it or not, it is possible to do Radio Astronomy at home. Especially if you consider Radio Astronomy to include all frequencies from "DC-to-Daylight". But even if your idea of Radio Astronomy is searching for deep space objects at microwave frequencies, you can put together a complete radio telescope for a few grand, assuming you can scavenge a Big Ugly Dish (BUD) from a neighbor who once used it to get Satellite TV.
You can also do interesting work at lower frequencies, probing changes in the ionosphere which are affected by both solar activity and also possibly by Gamma Ray Bursts from deep space
VHF and Above
The advent of inexpensive Software Defined Radio (SDR) USB dongles has completely changed the way amateurs do Radio Astronomy. Devices like the 
SDRplay rsp2
combined with free software allow you to receive signals from the Sun, Jupiter, or deep space.