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First and foremost, it's age and likely cosmetic condition. While some may immediately write these off as demerits, it frees you from parking half a mile away in the mall parking lot to avoid that first door ding and encourages you to do what that hunk tub of metal and gasoline was made for; driving and seeing what driving can show you.
In my roughly 10 years of driving (legally), I've owned 7 different cars and trucks and one of my favorites was a 1985 Volvo 240 sedan that I bought with 205k miles on the clock and a 4 speed manual with the optional push-button overdrive. Bought at a time in my life when all I could afford was basic transportation, it ended up taking me on the best road trip of my life...3 weeks, 6500 miles...without a single hiccup. I got it stuck in sand dunes in southern Utah while being laughed at by ORVers, drove through a June snowstorm across a mountain pass in Oregon and pounded up and down canyons in California, that 4-cyl grinding and growling its way across the country like an old diesel tractor. The 240 was no sports car but I had more fun flinging that thing around mountain roads than most other cars I've owned. Its deficiencies made it something easy to get the most out of on an every day basis and something you didn't really have to care about besides basic maintenance. It's 80's stereo even forced me to raid my dad's cassette collection to keep the tinny two speaker bumping.
Practically speaking, most older cars, if they were ever mass-produced, are cheap to repair and there are likely hundreds of them sitting in junk yards from which used parts can be pulled. Heck, you might even make a stab at doing some work yourself considering the engine bay in most older cars was simply a hunk of iron rather than 15 computers that throw 10 different warning lights at you every time you touch that wire.
Finally, the best thing about driving a beater is just that, driving. Broken records aside, driving a vehicle like that old Volvo forces you to pay attention to the task at hand. You can be going 10 mph but you feel intimately connected to everything that's happening...no computer mediating mechanical connections, choosing what it thinks you want to hear and feel.
The massive popularity of events like the 24 Hours of LeMons suggests I'm not the only person who feels this way. There's something particularly satisfying about wringing every last bit of performance from a car that was never built for more than weekend cruises down Main St. in 1965. When you don't have tens of thousands of dollars (or much more) invested in a brand new sports car, you're freed from the constant worry of seeing your pride and joy nicked or dented and you feel more free to really see what's its capable of.
When I bought my newest car, a 2000 Toyota Tacoma, which I'm currently outfitting to be a tiny camper, I ended up finding one in solid mechanical condition with a number of dents and scratches and I was thrilled! Since my plan was to have a vehicle I could explore the way-back roads of the Pacific Northwest in, I was going to get it banged up a bit anyways. With blemishes already in place, I feel free to go wherever she can take me, downed trees and narrow gaps be damned. Doesn't hurt I saved $2000 on the purchase because of those "blemishes" either.