First of all, I assume that you are using a working 45 W power adapter with a 24 V DC output (low-voltage plug: centre positive, outer ring negative).
The PowerBook 5300 has a standard problem with a loose DC low-voltage connector. If necessary, the connections to the board can be resoldered (details can be found through a web search).
Older computers may have issues with leaking electrolytic capacitors. A careful inspection of the board(s) should not show any signs of leaking/damaged/discoloured components, nor of cracked/bad solder joints.
If a faulty main battery is suspected (can put a heavy load on the circuits), one could check whether the PowerBook starts up with this battery removed.
Try to reset the power manager and to reset the PRAM.
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