Ashtama Siddhi refers to the kind of powers that is perhaps called Super powers by the West.

How does one get Ashtama Siddhi?
Definitely not an everyday affair. Physical control, coupled with mental discipline plus many secret ingredients perhaps.
Years of meditation, coupled with super human control of the body and mind seems to be the general consensus.
There could be certain individuals like some of our revered Gurus who may have attained some of these powers, although they may or may not want to disclose / talk about it. That is why we hear a few miraculous experiences with respect to our revered Gurus.

The Avengers version of super natural power seems quite easy in comparison.

What are these Ashtama Siddhis?
Anima – shrink at will;
Mahima – expand at will;
Garima – become heavy at will;
Laghima – become light at will;
Prapti – adapt to any environment at will (teleportation, under water existence etc);
Istva – ability  to control natural phenomenon at will
Vasitva – ability to control other minds including that of humans and animals at will
    
Hanuman is supposed to be the one who possesses Ashtama Siddhi. He has also been blessed with the power of conferring this on others who are fit to receive such blessings.

In various situations in the Ramayan, BajrangBali has exhibited these and that is perhaps why he is our own Super Hero