Fox,
I think people are getting the TDS300 series confused with the TDS400 series. The TDS400 series does indeed have a couple of batteries in it, the TDS300 series does not. At any rate, the state of the battery will not cause the unit to not power up.
Also, the TDS300 series was not plagued by bad electrolytics like the TDS400 and TDS500 series. While the TDS320/340/360/380 were built at about the same time, they didn't have the huge number of caps that the 400/500 series had and thus did not have near the problems. I've repaired dozens of TDS300, TDS400, and TDS500 series scopes, and while it might be a good preventative measure to replace the caps on the 300 series, in general they don't cause power on issues. Also, the TDS400 and TDS300 series of scopes do not use the same power supply. There is a schematic available for these on the Tek website though, that might help.
As others have pointed out doing repair of off-line power supplies is not a beginner project, and can be very tricky (and dangerous) to perform.
For what it's worth, I do have an old power supply from either a TDS320 or TDS340 that I'd be willing to let go pretty cheap, contact me off line if you're interested. Good luck!