LUKE: Jess, where did you get the money?
JESS: … Remember I work for you?
LUKE: … I don’t pay you enough to buy the car.
JESS: I saved up my pennies and I bought the car from Gypsy. She gave me a good deal. That’s how I got the car.
Luke has trouble understanding how Jess was able to buy a car. Lorelai helpfully suggested that Jess probably stole it, and with the uncanny ability Jess has to mind-meld with Lorelai, he also teases Luke by saying he mugged someone.
I’m not sure why this is all such a mystery. Jess’ car may be roadworthy, but it’s in poor condition otherwise, and Jess says Gypsy sold it to him at a bargain price. That could have been as low as $500, and is unlikely to be more than $1000.
Jess has been working at the diner every day for about a year, which was probably a full-time job over summer vacation, probably getting around $4 an hour. As little as 125 hours work could have paid for the car, and Jess, like Luke, has frugal habits, and is probably a good saver. (He also cleaned Lorelai’s gutters, and may have taken on other chores around town).
Even though there’s nothing at all unbelievable about Jess having an old car, it does get explained later in the episode.