Pilot light won't stay lit after repeatedly holding down that damn red button for 'a minute'
There's a good chance that the thermocouple is kaput.
You - as an average human with most of your digits - can safely swap out the components. Not easy, you need small hands or a big attitude... but after unscrewing the thermo/pilot assembly from the mounting point (I'd recommend getting those nifty palm size screwdrivers, cause a real screwdriver does not fit inside the access cave of a gas water heater), and unscrewing the fittings for the thermo/pilot assembly, you can then go to your nearest
The lesson, dear readers, is that if you're gonna waste a day at home fretting about the landlord visiting and noticing you've installed cabinets on the kitchen wall, or that the floors have seen better days (pre-blue chasing pepper and the ball about), or that you don't expect Mario to do much more than make things worse... empower yourself. Take control, do the reading on the 'net, and then get it done so you can move onto... animation of layers in Dreamweaver!
dona nobis pacem