There is a lot of that on the E-bay, it is called sniping, and it must work cause it happens all the time.
If you are a handy with a wrench kind of person you can cob together a rig for real cheap from any old frame with long horizonal dropouts.
42/15-17 ish seems to be a good starting gear. Whatever you do do have a front brake at minimum, I don’t care who you are a front brake will stop you faster and safer.
There are a bunch of new rigs for around a grand or look under “track” in the cycling section of the E-bay. you can probably get a rig for between $400-700 range. Be very careful it can be addicting, I look for TT with flat courses to use my fixie on all the time. Aloha G