$2k is the most I've ever spent on a car. If you buy right, you can get at least 2 years / 50k miles out of them (often more) with maybe only one or two semi-big ticket repairs (clutch, exhaust, etc.). We currently drive a 2001 Volvo V70 (bought at 140k for $1800, currently at 171k) and a 2005 Kia Spectra5 (bought at 113k for $2000, currently at 120k). Cheapest car was a 1992 Buick with 158k miles for $700. We drove it just over two years and 42,000 miles and the engine gave out just past the big 200k.
The only help-I'm-stranded breakdowns we've had on this plan since about 2012 were either not the car's fault (something blowing a hole in the fuel tank and spilling 18 gallons of gas onto I-94, flat tires due to running something over, broken aftermarket key in the ignition, etc), or the car giving up completely and dying. Only other breakdown instance I can think of was the serpentine belt pulley cracking off on the aforementioned 1992 Buick (that was fun).
You do have to be patient, know what to look for, be willing to wade through a bunch of garbage on Craigslist (lots of bad cars out there for cheap along with the good ones), know the quirks with a given model, and be willing to do a little work yourself (though it could still pencil out without DIY work if needed), but it's very doable.