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

Not only would you have to be extremely lucky to get a $3000 machine in good shape, you will have to get lucky finding good pre-recorded tapes.  Good tapes, that clearly beat out decent vinyl records can cost many hundreds of dollars each.  Blank tapes for recording costs $96 for a 2500 ft. reel.  This is an expensive niche in this hobby.

I second the Pioneer RT-707 or RT-701 (no auto reverse). Built like a tank. I've had a RT-707 in my garage system for 30 years.

before you plop down your cash on a tape deck you may want to consider what format you are most likely going to play. 1/4 track, half track, what speed? 7.5 ips, or 15 or 30 ips?

without determining these factors first pretty hard to make a machine recommendation. 

You can find good quality Pioneer, Sony, Akai and Teac decks for under $1,000 on CL.  Plan on spending an equal amount getting them refurbished.  It's not as expensive as some of the above replies state.

Yeah, you could get a good sounding deck for $1k but it may not be good enough. That's the main reason why I don't have reel-to-reel. I saw Otari MX-5050BII-2 both on ebay and on reverb for close to $3k in seemingly good or better condition. That's where I would start. Studer 810/807 is two/three times more.