Just to supplement Rory Alsop's answer: I've driven that road, and didn't think it was excessively steep or windy. I was in a pickup truck, but small cars on the road didn't seem to have any trouble. I routinely drive steeper and windier roads in a Honda Civic. I think any modern car in good working order will be just fine, even a compact or economy model.
(If you're not from the US, note that cars here are more powerful than in many other parts of the world. Even the smallest cars on the US market today have at least a 1.5 L engine producing well over 100 hp (75 kW). (Oh, sorry, your profile says California so you know this. But I'll leave it for the benefit of future readers.))
The major car rental agencies provide a pretty good product, in my experience. The cheaper classes of rental cars will be smaller but they won't be crappy (unless you're the kind of person that thinks anything smaller than a Lincoln Town Car is inherently crappy). "Tiny" is perhaps relative, though I expect you won't find anything smaller than a 4-door compact sedan, like a Toyota Yaris. Major rental agencies usually keep their cars only a short time and then sell them, so the car you get will likely be less than a couple years old, and less than 30,000 miles or so. Whenever I've rented, I've found the vehicles to be in perfect working order. There is a good chance they will try to convince you to upgrade to a larger vehicle, but there's no real need for you to do so.
If you don't have experience driving in mountains, you might look up some relevant driving tips before you go. The main ones are (1) slow down, and (2) downshift when going downhill to take advantage of engine braking and avoid overheating your regular brakes, which makes them not work. It's also wise to keep an eye on your temperature gauge when going up, as this increases load on your engine and increases the risk of overheating, but modern cars are much less subject to this than older vehicles.
(Let's see, you're from Long Beach. I've lived in Southern California too - I would say US-285 is a little more steep/windy than I-5 over the Grapevine, but considerably less so than the Angeles Crest Highway (CA-2)).
(Note: Based on your mention of rafting, I'm assuming your trip will be in the summer. If you're going in winter, my advice would change - you'd want to consider renting a vehicle with four-wheel drive, as that region gets a considerable amount of snow.)