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It sounds to me like the hard disk is not booting.
Using the original CD/DVD disks that came with the computer, insert the disk the one that includes Apple Diagnostics. Hold down the 'C' key when you power the system on and attempt to run diagnostics. If the computer shuts down before completing diagnostics, it is time to find an authorized repair center.
If the computer runs the diagnostics successfully and finds no other problems, it may be a problem with the hard drive.
If you don't have the original system disks, but have an OS X Snow Leopard (10.6.x) upgrade disk, that will still work to run Disk Utility. Hold down the 'C' key when you power the system on and select Disk Utility.
In the left column of Disk Utility, select the drive labeled 'Macintosh HD' (unless you renamed it.) Then select 'Verify Disk'. If it fails the verification, it is likely that the hard drive is failing or has failed, and needs to be replaced. If it passes verification, attempt to Repair Permissions.
If you have some spare time, you can attempt to reinstall OS X on the drive, and see if it will cure the boot problem.
You also mentioned that you thought the computer was "overheating". The automatic cooling fan runs at different speeds to cool the computer, and the hotter the computer is inside, the higher the fan speed and the louder the noise. Recharging the battery, editing video, editing photos and graphics, web browsing sites that use Adobe Flash, and even Twitter are among the activities that cause a lot of CPU and/or video chipset activity. That activity generates heat. The fan normally runs about 2000 rpm, but nearly silently. More heat causes the fans to run faster, up to about 6000 rpm. Over time, dust and dirt can also accumulate inside the computer, and heat dissipates less efficiently, causing the fan to run more frequently and/or faster.