Word of God answer:
The following is the last verse of the theme song. It was written by Joel Hodgson (the creator of MST3K, who also played Joel Robinson) and Josh Weinstein (founding member of the writing team, and the original voice of Tom Servo). This makes it a pretty compelling canonical answer.
If you're wondering how he eats and breathes
And other science facts,
Then just repeat to yourself "It's just a show,
I should really just relax
For Mystery Science Theater 3000."
In most seasons of the series, movies are transported to the Satellite of Love via a 227 mile long tube called the "Umbilicus". From the official FAQ:
Q: How could a tube run from an underground cave to an orbiting satellite?
A: It's just a show, you should really just relax. The concept was introduced to allow more interaction between Deep 13 and the SOL. For the record, the umbilicus/con/port was supposed to be 227 miles long. The writers at BBI once calculated that for an object to travel from Deep 13 to the SOL in 10 seconds, it would be traveling at 82,000 miles per hour!
Q: Why did Joel/Mike watch these bombs? What would have happened if he had refused to enter the theater?
A: In episode 208- Lost Continent Joel refuses to enter the theater, and is zapped by remote control by the Mads. So it appears that he has to watch them, or get a shock to the shammies. In episode 819- Invasion of the Neptune Men Mike is so repelled by the movie that he leaves the theater, only to be forced to return when he discovers that Pearl has turned off the oxygen supply to the rest of the ship (a trick that Dr. Forrester similarly used in MST3K: The Movie). So it appears that they had to watch them, or suffer the consequences. Besides, if they didn't, it wouldn't be much of a show, right?
Taken together, these facts create the possibility that the Satellite of Love is supplied with air through the tube used to send the movie reels to the Satellite of Love1. It would follow that food, and possibly water, were supplied in the same manner.
This is entirely logical: Dr. Forrester is evil, and his whole purpose in life is to torture Joel and Mike. He built the Satellite of Love for just this purpose - to torture its inhabitants. Thus, simple logic would dictate the following:
The satellite is in space.
The satellite exists for the purpose of keeping Joel and/or Mike captive and torturing them.
Humans can't survive in space without food, water, oxygen, heat, etc.
You can't torture dead people (although you are welcome to try, it won't bother the dead people, so it isn't really torture, it's just messing around with a corpse, which could be done far more easily if you and the corpse were both on earth).
Therefore, it is in Dr. Forrester's best interests to provide Joel/Mike with the basic essentials for life, not for their benefit, but for the purpose of being able to continue to torture them.
But again, it's just a show, so you should really just relax.
1 On a more fundamental level, it is obvious that, if Dr. Forrester is able to shut off the oxygen supply, an oxygen supply must exist.