You can purchase a home about a half mile near the best part of the Westside of Los Angeles for about $650,000 at 10% down with good credit (really nice neighborhoods west of Sepulveda). I haven't checked that out in about 6 months, but that's the reality nonetheless. So getting a home in LA (as just one example) is hardly unattainable for someone who has worked for, say, 5 years making $70K each year and saving, say, $15K each of those 5 years combined. You don't have to be in your 30's either, because if you saved smartly and started working straight out of college you can eventually get a salary like that (starting at less than $70K obviously) with that $15K saved up each year if you keep your expenses down initially so you can pad your savings later with a higher salary (also depends on your tax deductions of course among a slew of other choices like purchasing a car or not. I personally opted not to). So you can have a house in a nice part of LA by your late 20's without anything more than a college degree.
EDIT: I would add that purchasing a home in your 20's in LA probably isn't a good idea unless you're breaking 6 figures easily (or unless it's a killer deal with huge upside). Renting or even just staying at a family/friend's place is simply far more economical in your 20's than owning or even outright renting by yourself with no roommates to reduce monthly costs. Especially in big cities, it's just too expensive. But cities like LA, NY, Chicago, etc. are where businesses go and where the jobs are.