A variable power supply, DMM and function generator should be enough to verify the attenuation, time base and horizontal/vertical geometry. The only thing you can't verify/adjust is frequency response / transient response, which either needs a leveled sine wave oscillator or a pulse generator with fast, clean edges. In some cases it's only a performance verification anyway, it's either OK or it's not and it needs repair.
Much of the dedicated equipment is about doing the calibration faster and easier, which is a big deal for commercial calibration. If you want to follow the manufacturer's procedure, you're likely to need $1k+ worth of equipment if you buy used, and much more if you pay list price. But you should be able to do a reasonable job, at least for performance verification, with improvised equipment. I would be hesitant to adjust anything based on this. I would try to follow the standard procedure as much as possible.
A multimeter is in some ways harder, since accuracy is much more important. Even a very basic DMM will do 0.3%, much better than standard resistors or something like a 7805. AC voltage and current tend to be quite tricky to get accurate, especially at higher frequencies. Reasonably accurate resistance and DCV is achievable by spending a moderate amount of money (eg. the Geller and Voltagestandard products), and current can be derived from that. To do a full calibration/verification, you'll also need ways to generate the higher ranges, like 1000V and 10Mohm.