No. Game consoles, the original Xbox excepted, are almost entirely custom hardware. You might be able to pull the HDD out of the little case, and slap that into an enclosure, but that's about it.
The Xbox 360 uses a custom made PowerPC based CPU, and has a modified version of DirectX used as the API. You would need to emulate this on a PC, and that would require far more processing power than the average computer has these days. Maybe in 5-6 years, higher end computers might be able to do it.
What you may want to do however, is look into Microsoft's repair program. They have a flat rate of $100 ($120 if you do it over the phone) which is basically no questions asked. As long as you haven't done something that would void the warranty, they'll fix your console. Cheaper than buying a new one.http://support.xbox.com/en-US/xbox-360/repair/repair-process