A system board for your model (P/N P000285900) would cost $300 + and that is an exchange price meaning that you have to return your old system board. If you want to buy out right then looh for the price to double.
That price range reflects vendor mark up. If you know someone who has access to Toshiba parts direct and works for an easy-going repair facility (read - would allow employees to repair family/personal equipement without charge) then you could see a great reduction in price. Those prices do not include labor.
For your problem, try some trouble shooting. Remove all CRU's meaning battery, CD drive and floppy if possible or present (I think you have to crack the case to do that so don't try). Then attempt to power on the system with just the AC adapter.
If the sam, while powered "on" look for a "reset" button along one of the edges of the laptop. It should be marked "reset" and you will need a pen to press it.
What are the results?
If the same remove the hard drive and repeat the process, also try with a boot disk in the floppy drive.
If the same then it is most definatly a system board issue. Seeing how this a PII 400 (Celeron?) then you have got to ask is persuit of repair cost effective. If you want to try and do it yourself, then try looking for a system board on ebay then do it your self. If you go that route I might be able to get you step by step instructions on dis-assembly.
AL