Is there an 'Audiophile' Reel-to-Reel Machine for Under $1,000?

No.  And most likely anything you would find used and in need of repair would be a 4-track machine, whereas the new "audiophile" tapes are almost exclusively 2-track.  If you want a cheap machine, are you really willing to spend $400 on up PER ALBUM for the "audiophile" pre-recorded tapes?

The only decent machines that one can occasionally find for quite low a price is the Otari 5050 because it is a professional machine that was built in vast numbers and is extremely rugged.  But, because it is a professionally used machine, most of them on the market have been heavily used.  A friend who serviced machines likes them because of how rugged they are and they are relatively easy to fix.  He said that the basic build is even better than Studer machines.  The amount of work needed to get a machine really up to snuff is considerable, and aligning heads and calibrating machines requires a lot of skill and specialized gear and expensive alignment tapes, so it is not something that is a do-it-yourself project even if you are good at it. 

Many great sounding machines, like the Technics 1500 and 1700 models, are very much in demand and command high prices.  But even ones in good working order can be ticking timebombs.  The Technics machines are filled with cheap Panasonic electrolytic capacitors that WILL start to leak and the leaking electrolyte is highly corrosive and can destroy circuit boards.  That means anyone owning such machines should spend a lot for preventive parts replacement if they hope to keep those machines alive.  

home made: how many tapes? recorded lps to tape? recorded live broadcasts? copied friend's music?

what size reels, tape width, number of tracks, forward and/or reverse format, speed are the tapes?

pre-recorded: I assume there must also be some factory made, same questions ....?

shedding: what brand tape did he use? home made tapes might be the brands that eventually shed, the emulsion comes off the plastic strip, so do some research about that, which brands there is a way to bake tapes that shed

I love tape, it is my best sounding format, however mine is not awesome like my friend's 2 track 15 ips pro model/content

and the content is old, limited playI went thru several machines and settled on Teac, x1000r and x2000r, their last Pro-Sumer decks.

here's one in your budget you might ask more questions about, take a chance on

https://www.ebay.com/itm/306045178137?mkevt=1&mkcid=1&mkrid=711-53200-19255-0&campid=5338381866&toolid=10001&customid=38edd0f8-f9c5-11ef-ae90-636665636265

it plays 10" reels, it plays 1/4" wide tape; it plays 4 tracks (not 2 track stereo), consumer speeds 3-3/4 ips and 7-1/2 ips (does not play 15 ips),

you might consider finding someone who can play a random sample of them, see what they sound like, IOW there is a chance it was fun but they don't sound outstanding, or they are awesome, but so what, it is content you might never want to hear more than once.

IF they are 15 ips and awesome, you will need far more than $1,000, again, verify they sound awesome before spending a lot of money.

Old tape decks in working condition are good fun, and whether or not they qualify as "audiophile" they can sound GREAT. Very warm, liquid sound quality. The antithesis of dry, analytical hifi sound. It’s a lot of fun. You may notice the rolloff in treble, depending on what tape & deck you’re running. For my older decks, I found that adding Tannoy superwteeters helped with this, even as they were "too much" for my vinyl and CDs.

I had a Pioneer 1020L that sounded GREAT until a channel went out. They should still be findable, in good order, under 1000.

I now have a Pioneer RT909 that also sounds great, but probably not better than the cheaper 1020. These cost more than 1000, but should be findable at or under 2000. The RT707 mentioned above should be a great option; its limitation being the smaller reel size. There are other, less desirable 1970's R2R decks I've heard hooked up, and they sounded great too! One blew a cap WHILE we were enjoying it so...that's a risk. 

The decks I’ve had were quarter track (2-sided stereo), quarter-inch tape with top speed 7.5 IPS. I don’t have any 15 IPS tapes. The 7.5 IPS sounds really good - even being needledrop tapes from another audiophile. Below 7.5 IPS is where it loses the magic for me. High end / audiophile stuff is typically 15 IPS.

The stock output levels of these tapes decks is quite low. So it helps to have a good preamp with some extra gain to spare.

The audiophile / studio decks and tapes move up to half-track (one-side stereo), 15 IPS, etc. That’s a lot more tape, and audiophile tape issues are obnoxiously expensive ($400+ a tape), and extremely limited in content. They sound amazing, though! The new tape decks are very expensive, understandably so.

I think, we made it sound very complicated to the OP. But, yes, he must decide if he needs two or four track deck and if he needs 15ips speed.  Master tape dubs can be less than $400 but still very expensive. For his old recorded tapes he may need 3.75ips speed, we don't know. For less expensive deck than Otari I would probably go after either Pioneer or Revox.