Hi blobby,
In addition to Corrine's suggestions:
What is the make and model of this laptop?
Is the battery charged? Have you tried booting with the battery only?
I have seen a similar problem with a Toshiba laptop. It turned out to be a faulty battery. Try removing the battery and power it up using the main adapter.
Otherwise, sounds like a possible Mother Board issue which can cause intermittent shut downs or failure to start.
Also, if you hold the power switch in for about 10/20 seconds it might perform a soft reset.
How old is this laptop? Is it still under warranty by any chance?
how can I retrieve photos etc that I haven't already backed up?
Only thing that comes to mind would be to slave the drive.
Personally, I would not recommend opening any laptops as they are very compact and it is difficult to do work inside them unless you have previous experience.
The cost of a new Mother Board is about a 1/3 the cost of a halfway decent new laptop if money is an issue. A reputable Tech. Shop can perform a diagnostics test on the Mother Board and other Hardware and can retrieve the info for you safely.