Welcome to SailNet. The C27 is a remarkable boat for the price, with 6000 produced and still sailing. They probably can't handle the worst conditions, but they're easy to sail, fairly fast, and best of all, fairly easy to maintain.
When buying a boat, really the most important things are the condition of the hull and rigging, and the condition of the motor.
So, the general concerns I've heard about are insufficient support for shrouds, and deck delamination. Walk around on the deck and check for spongy or squishy spots. Repair of this problem is challenging and involves not sailing for a while. There's lots of related info on this forum. The upper shrouds should be fastened to chainplates which run through sealed holes in the side decks and bolt with many many bolts to the bulkheads. Older C27s had the upper shrouds bolted through the side decks to backing plates, like the lowers. I have no idea how difficult it would be upgrade the older system, though I believe Catalina used to sell kits.
As for the motor, if it's old there's probably a lot wrong with it, unless the PO can give you evidence of maintenance. Don't assume that just because the PO or broker started it up for you, it will start up when you try it yourself. Motors are remarkably reliable -- if properly cared for. It's hard to check the condition of a carburetor, impeller, or starter motor with just a ten-second start and run test. On the other hand, a bad motor is not expensive to replace (I used a $1000 4 hp last season while the 10 hp was out of commission) or repair, if you can do the work yourself.
One of the things I'm not happy with on my boat is the state of the wiring, which should be neatly tucked away but in my case is sort of... hanging around everywhere. Fortunately this is something you can also fix yourself.
More things you can fix yourself: worn rigging (buy a splicing kit and learn to use it), minor sail damage (anybody can sew), tiny pockmarks in the gelcoat (spinnaker poles and anchors on deck will turn your gelcoat into swiss cheese; not structurally problematic if dealt with early, and not a job that you need a professional for).
Just because the price seems too good to be true, doesn't mean it is, especially nowadays. Just don't let the juicy accessories (like sails) distract you from the important stuff (hull and motor). Also, use that money you're saving to get a thorough surveyor.
1972 Catalina 27