If you allowed covers (and why not?) then certainly, and you could likely even have a single type of music.
Indefinitely? With no repeats? Nope.
Sooner or later some kind of mistake will occur.
For example, you’d probably use some kind of database to keep track of what’s already been played in order to prevent repeats. At some point two versions of the same song with slightly different titles will come along. You’ll play the first one and dutifully record it in the database. Later - maybe years later - the second one will come along and it’s title won’t match anything in the database and you’ll play it. You’ve then failed in your mission. You may not realize it, and you may continue on obliviously, but you’ve failed.
That’s just one example. All kinds of mistakes can be made and, given enough time, sooner or later one will be. The sheer amount of material is as much a negative as it is a positive.
It’s been noted above that enough new music is being produced to maintain airplay without relying on the back catalog. Instead, the back catalog could be used as a buffer of sorts, allowing for decades-long, absurdabytes’ worth of error checking. The catalog could also be used to liberally err on the side of caution–too many notes, chord changes, or lyrics sound the same? Dump it, just to be on the safe side.
So far as the amount of music being produced goes, allmusic lists over 400 new releases for this week. Some of them are re-issues, but there’s still a massive amount of new music being put out every week.
When it runs out of money.
It is doutbless that you can broadcast 40 hours of no repeat music every week.
Whether you can broadcast 40 hours of no repeat music every week that enough people want to listen to, is a very different question.
If not enough people listen to it, you won’t have the subscription or advertising revenue you need to keep it running although you might be able to run it as a non profit.
Perhaps you could get enough music for free if you set it up as a way for artists to get publicity.
There’s always banging a hammer against a trash can through the whole show. Call it experimental music and get a grant.
Yeah, I just never thought about it, really. I mean, we probably hit the ‘literary singularity’ a couple of hundred years after Gutenberg… Oddly enough, I recently read a short story that touched on this point (it’s available online here), but somehow, it just didn’t really click until now.
I still think that given enough time something will slip through. For one thing, the definition of the concept “same song” is fuzzy. Same music, different lyrics? Same lyrics, different music?
Not only that, but the time to do the searches will increase over time. Yes, I know that with a properly designed database, title searches shouldn’t increase in time too quickly, but when you talk about comparing lyrics, notes, chords, etc. then the more songs you’ve played the longer the search will take. If we’re talking about doing this indefinitely then at some point the catalog of already played music that has to be searched will be so big that the search on one “new” song will take so long that before it’s finished you’ll run out of already vetted material to play. Unless, of course, you’re willing to use more and more (and more) of the universe’s resources to store data and conduct searches in parallel.
This question is really about information theory.
Another issue. Assume that there is some maximum length to what songs we’ll play. (Without this, your data problems would be even worse, among other things.) Say we won’t play anything over X minutes. There are a huge, but FINITE number of combinations of human perceivable sounds of length X. Assuming we want the “songs” to actually be recognizable as some form of musical entertainment, as opposed to random noise or long rambling incoherent (or even coherent) speeches, we’re limited even farther. You’ll eventually play all possible “music” of length X or less and run out of stuff to play.
You can’t do this forever.
Even if we can imagine perfect machines with perfect software, some human error will occur eventually.
Even if we assume that somehow there will be perfect record keeping and vetting of songs, the length of time to vet a song will eventually become too long.
Even if we assume that we somehow overcome all of those things, we will eventually (granted, after a long, LONG time) have played all possible songs (or at least all possible songs that meet our criteria).
Of course, long before any of that the universe will have ended. 
The universe ending is a key point. A while back there was a great thread about a monitor displaying random pixels and therefore showing every known and conceivable picture. I don’t recall the math, but the universe would have collapsed into itself several times over before you got a nice image of monkeys on Shakespearian typewriters.
Similarly trivial is the notion that any system is subject to a mathematical, mechanical, or some other physical error. Do we currently have the technology to construct a machine to count by ones forever? Saying that we don’t because it will inevitably produce some random error doesn’t really address the question–not that it’s not a good question in itself; see the recent thread about the feasibility of sending a device to another star.
But in this instance, the enormous backlog of music creates enough space to work with in checking each song for a set of pre-defined parameters (voice recognition for lyrics, (in multiple languages), pitch/tempo/beat analysis, etc.). This is (relatively) trivial computing, and will remain so for quite some time–and as the playlist grows, so will computing power.
So be sure to pick a really good song to usher in the end of the universe.
Hotel California perhaps? 
I read once that after Armageddon, when only cockroaches remain, there will be a shutter banging in the wind and an old battery-powered tape deck playing Hotel California.
I’d like to suggest rejecting all versions of The Hollow Menset to music, saving it for the final moment.
WRDU did something like this in 1993. The morning hosts decided they were going to play every single song in the station’s catalog in alphabetical order. It caught on with the other shifts. I think it took ten days to play everything.
I talked to one of them at a live event. He said they initially wanted to play everything alpha-by-group, but they would have had to sit through the ABBA set.