It would take some downtime (or at least a really long extension cord and redundant power supplies for swapping one-by-one) because you'd still need to connect it to power.
The solution is obvious: Qi inductive wireless charge coils on the outside of the server cases.
You swap it over to a desktop UPS that can come along with the server during the physical move. No downtime as long as you have redundant PSUs and you can walk fast enough to beat the battery draining.
VW makes an excellent 2.0L TDI that would be a good fit. It would take up about half a rack. You could integrate a fuel cell (say 5 gallons) so that it can function when it is detached from the fuel delivery fans.
If you don't want to worry about piping diesel around your server farm, we could go with compressed air and use air turbines with generators on them. Clean and efficient energy transmission without wires!
The idea of having a miniscule diesel generator in the server chassis is almost worth doing it right now. Still have fond memories of our miniature steam engine, but I figure that would be a bit impractical.
The solution is obvious: Qi inductive wireless charge coils on the outside of the server cases.