OK.
Hardware lockups are caused almost all of the time by one of two things. Power or heat.
No battery is not a problem. If you pull the plug out while it is running, and it shuts off will prove this.
Unfortunately, not having a working battery makes power diagnostics a little trickier. If it would hold a charge, you could easily eliminate the AC/DC adapter by running on battery only.
Adapters can be flaky since they are usually cheaply made, and voltage drops without a battery will cause the system to lock. Putting a load on them and checking voltage/amp draw can weed out a bad one
Again, if you have NO battery power, then as noted above, check the power connector on the book to see if it wiggles. DELL is notorious for having these contacts break or cold-solder.
Heat - notebooks are notorious for sucking in lint and plugging up the metal heatsink located over the CPU and other hot running chips. This will cause the CPU temperature to rise, and once it hits it's threshold it may alarm (if set to in the BIOS), then shutoff to protect itself. As a byproduct of these overheating cycles, the thermal paste that is put between the CPU and the heatsink will dry, crack, ooze, reducing it's ability to efficiently transfer the CPU heat away, again causing overheating.
My guess based on the age, would be the lint buildup. Flip over the book, remove the myriad of tiny Phillips screws, remove the back cover. If its plugged, clean it, fire it up, and ensure the fans are working properly. Thermal paste can be found at a real computer store, and is cheap. If you wnat, unscrew the cooler, put a thin coating on the CPU die (the metal portion) then reassemble, test.
Less likely but possible, it could also be a bad piece of memory (RAM). This can be tested by using free software like memtest, or removing one piece at a time in order to eliminate the bad one.
Hard Disk? Possible, but not from the sounds of it.
Of course it could be more serious, such as a cracked mainboard, or possibly a defective video chip (nVidias have had problems). In this case, fixing it is not feasible.
Grant