Sending an (old, generic) oscilloscope out for calibration is usually only done when some regulation or written procedure demands it. It is pointless for the normal user. The front panel calibrator is not relevant--it is for compensating the probes. If you want to do a check and adjustment, which is what 'calibration' should really be called in this case, you just need a decent signal generator or two, a DC PSU and a DMM that is not broken to get started. The reasoning here is that the scope is like an analog VOM--it is only accurate to 2-3% at the very best both vertically and horizontally. Calibration is making sure it isn't broken and adjusting it a bit here and there to get the picture as good as possible. I haven't seen that exact model, but on those later-model CRTs, most of the actual manual adjustments will be in the CRT section ad the rest will be automated. So I would keep the money in pocket and look at acquiring what you need to test it. Also, read the manual and figure out how the memory is retained. I think it is EAROM, which is battery-free, but I'm not sure.
As for the HP3478A, if the calibration memory battery has not gone bad, then it is likely still pretty accurate except perhaps for the low-ohms range. So the first order of business with that, IMO, would be to read up on how to carefully replace that battery without disturbing the memory. It is a bit tricky and there are a couple of ways to do it, but one common way is using an isolated soldering iron while the unit is powered up. You can also use an auxiliary battery with separate leads. Great care is needed no matter how you do it. Once that is handled, get something like the DMM Check just to verify that it hasn't lost its memory, which would cause an error big enough to notice. If you want to fully calibrate it, you need a lot more than a single source. Read the manual to see what all the required stimuli are. And definitely don't bother calibrating it, at home or by sending it out, without a new battery in place.