I worked at an auto auction for few years in my younger days, typically "older" cars owned/driven by retired folks tend to be well maintained, free of accident related damages/neglect, and have clean title.
There're plenty of good choices for $5000, so don't get discouraged. But one must do his home work & research to find that diamond in the rough.
On top of the basic mechanical inspections, I must emphasize that
"Vehicle History" is critical, undisclosed accidents, frame damage, tampered odometer, failure to pass smog check, flood dmg etc. will all bring surprises down the road. Some insurance company don't touch cars with Salvaged Title which can be caused by a variety of reason from frame dmg related collision to theft.
Make sure VIN numbers match up on all panels of the car, if there's VIN number stamped on driver's frt quarter panel under the hood but not on passenger side, then you know it has been hit & replaced.
Inspect the bolt & nuts on door hinges for tool marks, it will reveal whether the bolts have been "turned", a sign that door has been replaced.
In my state seller is responsible for smog check a vehicle in private party sales, I'd make sure the seller do the smog & not you. Do not ever get stuck w/ a car that can not pass smog.
Keep in mind, most folks do not want to deal w/ the hassle associated w/ private sales, instead they take the path of least resistance & trade-in at dealers or sell it to CarMax.
But they seldom get what they wish for at a dealer, a $7000 car could be worth only $5000 as a trade-in, consequently they are back to square one, sell it privately on Auto Trader, local paper or Craiglist etc.
They are out there.