I donno.
An age old question...and I'm not sure I agree with all of the answers. I don't intend to get into a "mine is bigger than yours" contest. But...
Yes, the chassis can certainly handle the horsepower. There are lots of examples of cars running near or more than 650 bhp. So, no, the chassis won't twist into a pretzel or anything like that.
Can you put the power to the ground? Can you use the power even if you could? Those are two separate questions, and the answer, IMHO, is it depends.
Putting the power to the ground depends on your suspension set up, your tires, your gearing, and I suppose your clutch. I have a track suspension set up, very tall gearing (in theory good for 210+ mph), and make 601 bhp at the flywheel. Do I have enough traction to put it to the pavement....yes, mostly. Out of first gear, and pointed mostly straight, I can stomp on it and not worry too much about losing traction.
Now, can I handle all of that power. Well, I'm not dead, so I suppose that the answer is yes. Realistically, I can't put the pedal to the floor for long when I'm on the street... no matter what gear I'm in (outside of first...good for about 65 mph). But I certainly wouldn't want less power.
I'd have to say that 350-400 bhp is NOT enough for anyone who has any experience in relatively high horsepower cars. It's just not. If you're used to driving a Honda civic, 350-400 bhp might be fun for a while on the street. But I think you'd get bored pretty quickly.
Contrary to what you might think, it might be enough for most people on the track where you can keep pretty good momentum going, and would be able to hit triple digit speeds rather often. Conversely, on the street, I find myself getting most of my thrills during rapid acceleration, rather than running at 100+ mph with any frequency. Hence, the need for horsepower. Besides, who want's to get smoked by any number of modern econo-boxes in the stoplight derby? I think my lawnmower makes 350 horsepower...and I'm sure that there are several minivans that do, too. Talk about being embarrassed!
So. How much is enough? I don't know, but I have a friend with a Kirkham that makes 750 bhp and has fairly standard Cobra gearing...I think he has enough. You so much as breath hard on the laud pedal, and your head gets snapped back. I've seen the car break traction while accelerating at 60-70 mph, and I doubt that the driver really had his foot in it. I briefly drove it, and it was scary fun.
In the final analysis, yes, the chassis can handle the power. Can you?....probably. I'll take flack for this, but unless you really just want to drive around and look cool, I wouldn't build one with less than 400 bhp.
What was that old Mark Donahue quote....?