I would like to dig a trench/ditch 10 feet deep, 5-10 feet wide, 35-50 feet
long. The space between the neighbor's fence and my air conditioner is as
little as 6 feet wide, so there is a limit to the space available to store
dug-up dirt.

So, my questions a

1. What kind of equipment should I get? Backhoe, trencher?
Bucket width if backhoe? Since my width is as little as 6 feet, I can't
have too much dug-up dirt to deal with.
I understand trenchers only go down about 30 inches and have wheels 5-6 feet
wide, so does the dug-up dirt go outside that width?
Should I rent the equipment or engage a contractor to run the equipment?

2. Any idea of the cost? I'm hoping for a few hundred dollars tops.

Here's what I'm doing, if you care to know: The terrain slopes toward our
house. The house has had water problems for years. Water has swirled
through the crawlspace and it looks like some of the supports have been
replaced and some rotted crossbeams have been replaced. The crawlspace has
been enclosed with concrete blocks, obviously after some of these other
problems have been fixed. I found that they were not properly waterproofed
and had holes caused by huge roots from a neighbor's tree, which seem to
have been undetected because they were hidden behind the gas meter and some
enormous azalea bushes. I dug around the outside to a foot or two below the
concrete blocks, waterproofing with stucco paint (stretches to 6X) and a
6-mil vinyl liner. So far, all this seems to have completely eliminated
moisture in the crawlspace.

However, in working on the back corner of the foundation at the end of the
waterproofing project, I dug to 2 feet and got some moisture. Dug to 4 feet
and I had a pond. The top of the water would still only come to 2 feet
below the surface. No matter how furiously I tried to empty the water, it
would maintain that level at 2 feet below the surface. Once it began to
seep, it continually increased in flow rate. Downhill, the soil under the
deck was always wet; I never knew why until now. And the door below the
deck began rubbing on the deck above it recently, so the deck is sinking. I
dug four-foot holes at various places uphill from the original point of
discovery and found that the water level is always 2 feet below the surface.
Understand that I don't know how deep this water table is. But I think the
house will be fine if I can keep any groundwater/water table at least a
couple of feet below the foundation.

So, my plan is to dig this trench 10 feet deep. Since I expect the trench
is going to fill up with water to 2 feet below the surface, I would attach
metal fence posts to 10 foot wide 6 mil vinly sheeting to weigh it down and
pull it to the bottom of the water-filled trench. Now, I have this vinyl
going to the bottom of the trench; but the seeping water might build up at
the vinyl sheeting and try to find its way around it. To prevent this, I
would attach a 4-inch slotted pipe with sock (as used in french drains) at
about 5 feet down the 10-foot sheet to carry the built-up water downhill to
a leech field. I would staple the top of the 10-foot sheet to old wood that
would hold the sheet up while I filled in the trench. If I have to use a
trencher (and thus gamble that the 30-inch cut will be enough to carry the
groundwater away), I would modify the size of the sheeting and the placement
of the pipe. I don't see any point in hauling stone in for the trench; the
ground is quite pourous and the stone thing would introduce a whole new
world of installation problems. Anyway, with a few days' work, I think I
can satisfactorily control the problem.