I'm going to go with the narrator as command module of the Apollo 11 mission, singing to the lunar module. Possibly to an astronaut who landed, but I think the LM:
The perfect fall: orbit
It was killing me: using up fuel
You were just an astronaut: LM inside, along for the ride on launch
Heat against your skin: the bright metallic skin of the LM
Hear the things they say: radio comminucation
Go without me: LM landing alone after separation in lunar orbit
Setting up the sea: Sea of Tranquility
Everyone is with you: hmm, doesn't fit. Anders stayed in the command module
Haven't caught me yet: Not yet re-docked for the flight back
I'm going to go with the narrator as command module of the Apollo 11 mission, singing to the lunar module. Possibly to an astronaut who landed, but I think the LM: