Most of the obvious ideas have been hashed though, a couple thoughts that
may have fallen through the cracks...

Garage door openers have been mentioned. Some are screw drive and would
likely be difficult to extend, but a complete screw replacement may be
possible. One of my suppliers is a company called Roton, they roll-form
Acme screws every day and three meters would be nothing special to them.
(over 10 meters the shipping becomes difficult, not the manufacture) So
it's at least possible... (still doesn't make it a good idea :-)
My garage door opener uses a timing belt. Open ended belting is available
by the roll, so again length would be a non issue here.

The sliding gate openers I have been around have a length of roller chain
stretched from one end to the other of the moving portion. A stationary
unit with a gearmotor has a matching sprocket and a couple idlers. A minute
with google gave me this link
http://www.wholesalegateopener.com/ALEKO-AC2200-Sliding-Gate-Opener-p/ac220=
0nor-wg.htm

That may not be an exact fit for your application as it seems you'd have to
stretch the chain on the stationary part and mount the unit to the moving
portion. But it seems a relatively straightforward "hack". Might also add
one or two more idler gears and run the chain "down and back" allowing you
to stationary mount the power unit.

Some older residential garage door openers also used chain, so I suspect
you could adapt it to this type of mechanism as well, may be mode of a
project than you're looking for.

Both of these ideas have been used for "low cost" homebrew CNC machines, a
quick google gives me
http://i1031.photobucket.com/albums/y380/gene8522/IMG_4946.jpg
(search for "blacktoe CNC" for more examples)

I've recently been working through 3D printer stuff and there's an idea
there you could adapt:
http://4.bp.blogspot.com/-FcyQhXc1EFs/Uh0dE01_nPI/AAAAAAAADSw/9NRCK2yJGjI/s=
1600/3DR_2_Motor_spool_wound.jpg
They're driving the printer parts with fishing line wrapped around a spool
(and achieving sub-millimeter repeat-ability and accuracy)
I suspect the idea could easily be scaled up to something with aircraft
cable and multi-meter travel. Depends on finding a suitable motor to drive
it...  The "secret sauce" to this one is in how they build the cable spool
and attach the cable to it. Rather than a capstan with "a couple wraps"
(which will eventually slip, a little at least) The center of the cable is
locked to the drum and does not move. Then enough cable is wound onto the
drum to permit operation - (poor explanation, read through the pictures.)

Given your speed requirements, I could see this as a stepper motor (larger,
plus gears) giving you "move to X" ability. If you find that interesting, I
can point you to some suitable drive electronics and motor options.

-Denny
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