I have had the same problem at one stage and before u try modchips or anything else i would just try a few different brands of dvd to boot from. An easy way to test if this will likely work is to put a game disc in, reboot ur xbox, and see if u can play it. If u cant play the original game disk then you might have to go to a more risky approach.
If it does then all u need is a dashboard boot disk to get ftp access to ur box.
If ur xbox is still having issues reading these boot disks then slayers might work. Sorry i cant say much about slayers cause i really dont know about it.
My next step would be using the eeprom backup (if u have one) to unlock and then use XboxHDM to move the files around. Or just building a HDD from scratch.
If u dont have the eeprom then u can still get XboxHDM to read the xbox hard drive by "hot swapping". You can do a search any easily find a tutorial thatll tell u how. Caution: if using the hotswap method, some tutorials that i have encountered do not mention that u should connect the PC case to the xbox case. Not connecting them could lead to electric shock and more importantly, frying ur xbox and PC
. This is a simple fix. Before turning on both the PC or the Xbox just get a piece of insulated wire and connect (stickytape, alligator clips etc.) one end of the wire to the metal case on ur PC and the other to the Metal case on ur xbox. Easy
hotswapping may be ur last resort before going out and buying a modchip.