After a few more attempts it suddenly decided to turn on properly. Only now the image is much brighter than it should be at the brightness setting it's at. I had to pull the brightness slider all the way down just to stop the blacks from looking too washed out. Also, when the on-screen image is dark, there are several thin diagonal blue lines across it, like this crude artist's impression:

This becomes less pronounced the longer the TV is left on, but fades back in periodically. Based on these symptoms, does this sound like something that could be repaired without too much expense, or is the CRT itself going?
EDIT: Removed the comment about this not happening in RGB mode, as it's got the blue lines in that now too.