An E65 error on boot of an Xbox 360 console represents a general hardware failure. A user will be prompted to contact Microsoft Customer Support with an on-screen error message. Additionally, a single blinking red light will be present in section 4 of the ring of light on the front of the console, as can be seen in the picture.
An E65 error is similar to an E64 error in that it usually has something to do with the DVD-ROM drive. One known cause is the drive not being DMA configured. This could be a result of tampering on the part of the user (in which case, your warranty is gone). However, in some cases, this could simply be caused by the DVD-ROM drive tray not being fully closed when the console boots.
This error is also related to a problem with firmware. This could be a result of a user attempting to flash modified firmware and being unsuccessful, in which case, again, the warranty is gone (if the console had any to begin with).
If this error is seen after flashing, then attempting to restore the original firmware might be a good idea. If that fails, another possible option would be to install a replacement drive spoofed with the original drive's DVD KEY, which the user should still have from the flashing procedure. A Samsung drive would be particularly useful here, but would have to be spoofed to the make and model of the original drive (unless the original drive was the same). Errors in flashing might also be caused by "incompatible" SATA controllers on the computer used to flash.
The E65 error is associated with an error code of 1001. To get an error code from an Xbox 360 console that is displaying an E65 error (or 3ROL) you simply press and hold the "sync up" button (used to connect wireless controllers) and then press the Eject button for the DVD drive. This will bring up the first digit of the error code (usually it is one light blinking red, which represents 1, please note that four lights blinking red represents a 0, not 4). With the sync up button still held, press the Eject button again and you get the second digit. Do this twice more to get the full 4 digits of the error code.
For more information on the error code you receive, or on Xbox 360 hardware problems in general, read this Xbox-Scene forum thread.