Am I reading the datasheets too strictly? Do you just need to make sure the selected input that you're asking it to pass through is less than the supply voltage, but can allow higher voltages on the unselected inputs?
No, there are going to be diodes from the input pins to Vcc. Exceeding Vcc by more than one diode drop (~0.6 V) will forward bias these diodes. The best case is that they start conducting and load the input. The worst case is that they blow up. Assuming there are protection diodes, a large value series resistor will limit the current through this diode to safe levels. Otherwise you need to provide clamping diodes yourself. If one of the analog switches is closed to provide the correct division ratio, than the clamping diodes should not be conducting, since the voltage will be below Vcc.
So... how do multimeters do it? I get that they use a resistor ladder with multiple taps, but what are they using to select the tap that goes to the ADC?
One technique is to have a high value resistor (eg. 9 Mohm) on connected to the positive input terminal (assuming it shares ground with the measurement circuit), connect this to one side of all analog switches, and then connect the other side to various resistors. The 9Mohm resistor would limit currents to safe values at all reasonable input voltages. The resistor at the other end of the analog switch would be chosen to get the correct division ratio (eg. 1 Mohm for 1/10x, 9 Mohm for 1 x, 90 Mohm for 10x). You can look at for example the
Fluke 87 service manual to get an idea of what they did. You might be able to do the same with a digital pot (I've never bothered with these) plus clamping diodes.
Note that you could go much lower than 9Mohm if input impedance is not an issue, especially for 30 V inputs. Just make sure the current through the (internal or external) clamping diodes does not exceed the maximum current of those devices.
Or is there a completely different approach I should be using?
If costs and size is no object, there's always relays. That's what many bench meters use for at least some of the ranges, since they provide excellent isolation and can easily switch voltages much higher than the voltage across the coil. Another technique is dividing the input down so even the highest range is below Vcc, and amplifying the signal between mux and ADC.