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I looked through a few inductors' datasheets and in none of them is the maximum voltage specified. Even Mouser does not let you filter inductors by voltage. Why is that? Is inductor voltage "not important"? How could I know, what voltage can be applied to this inductor?
I need >400V inductor, because I am building a boost converter. I already have reached >70V, but I stopped because of given reason.
If the inductor voltage is not specified, you can assume it's limited to 50 V or thereabouts of standard enameled copper wire.
An inductor that can handle the voltage you want will be explicitly specified for it. Look at some more data sheets, or contact the manufacturers directly.
Depending on the number of parts you want, you could also consider winding your own, using wire with an appropriately specified insulation.
How could I know, what voltage can be applied to "this" inductor?
If you are talking about the voltage across the inductor then you need to factor in the frequency of the applied voltage and the maximum current that the data sheet states.
$$V = L \dfrac{di}{dt}$$
So, do a little maths and work out what this voltage might be given the operating frequency (and therefore a strong hint to \$\frac{di}{dt}\$). You should also consider that this type of inductor will not be recommended for above a certain frequency (see self resonance) and this further constrains you in \$\frac{di}{dt}\$. If the self-resonance frequency isn't specified (or derivable) then my rule is (and I firmly stand by it) is don't use it.
And if you are still intent on using it and don't have the data, speak with the supplier.
You likely don't have 400 V between the terminals of the inductor; perhaps you have 400 V between any terminal and another part of your circuit (ground ?).
The high voltage insulation capability of these inductors is not specified; in high voltage applications you will need to ensure through spacing and perhaps additional insulation that your specification (and safety requirements) are met. You must assume negligible performance of the plastic protection wrapper on the inductor.
