during the turn-on (short), the current can be as high as 128A. 5V/0.039 = 128A. How is it possible for that MOSFET to handle that over time?
This won't happen because the LM3478's current sense input has a threshold of 0.156 volts and, you have a 39 mΩ resistor for Isense (R12). Simple math reveals that the MOSFET peak current is 4.00 amps but, your inductor will saturate at 3.6 amps. Data sheet extract: -

In my boost converter I am seeing the inductor burn after some uses.
This will cause the inductor to get quite hot and may indeed fail. The MOSFET may get a little warm too. Personally I think you need a bigger inductor; one that doesn't saturate until 5 amps and can handle the RMS current passing through it.
That RMS current cannot be determined with the details you have given but, fixing the inductor saturation current may be good enough. Also take note of the Irms testing box in the data sheet extract above; they tested the inductor with a fair amount of copper (which will act as a heatsink).
More detail from the OP
The OP revealed that the load current is 1 amp and, looking at his circuit, it looks like it will be operating at 500 kHz. So, with an input voltage of 5 volts and an expected output voltage of 26 volts (1 amp) we can expect an inductor RMS current of greater than 5.2 amps and, considerably greater than the 2.5 amp limit: -

Image taken from my basic website. The image is a calculator for boost converters but, it assumes ideal components for the inductor, diode and, MOSFET. I fully expect that the RMS current may even approach 5.6 amps in a real circuit.
Also think about this: the DC resistance of the inductor is typically 0.091 Ω (see data sheet extract above) so, the \$I^2R\$ losses will be about \$5.6^2 \times 0.091\$ = 2.85 watts so, no wonder it gets hot!
Elephant in the room
The other problem is this: with a 4 amp current limit, the output of 26 volts and 1 amp (26 watts) can never be met. Think about it; if the input power is (at best) 5 volts and 4 amps (a power of 20 watts) then how can you possibly get 26 watts to the output?