(1) Collecting stamps is my hobby.
(2) My hobby is to collect stamps.
(3) My hobby is collecting stamps.
(4) My hobby is that I like to collect stamps.
Do any of my sentences sound natural to native speakers?
All of the sentences except (4) are OK. Sentence (1) and (3) are transforms of each other, just as a transform of (2) would be "To collect stamps is my hobby."
Sentence (4) doesn't work because your hobby does not itself involve liking. It involves simply the doing of it. I recommend breaking (4) into two sentences:
(4a) I like to collect stamps. It's my hobby.
"It" in "It's my hobby" refers, of course, to the infinitive "to collect stamps," which again yields "To collect stamps is my hobby," by itself an undesirable construct.