Make it into a nice rock garden, before you chop off a foot!
Knowing of which i speak, altho i lost my foot in war, there are certain draw-backs to doing a lot of things!
for one--wading in the ocean is next to impossible-think about it--!
Two, if ya lose the right one, it means having to change the gas pedal on your vehicles from right to left side of the operators position!
Three--long hikes aren't recommended, without a backup driver to pick up yer bones and take ya home!
Four: Make sure you have a dependable vehicle when ya go out on a day-long fishing trip to yer favorite spot! Walking home is not an option! And, wading in bare feet is definitly a no-no!
And, rocks don't need mowed!

how steep is steep? you will likely need weights and chains or ags whatever ride on machine you get. if it is steep enough that you sldie with teh rear wheels locked, it may be too steep for low end machines unless you get a gear driven tranny instead of a hydro.

if it is not too large, why not use a push mower on this one area?

It is real steep and real long. I have a guy that cuts it now (just the hill, I cut the rest of the yard) with a wide cut zero turn mower. But I would like to cut it myself(one of those challange things, I guess.)

I can do pretty well on the old Torro hydro except in the center, which is the steepest part. That part is probably 60 to 80 feet long and maybe, like 60 degrees.

It's way too much for me to mow with a push and it would cost a small fortune to rock it, or something similar. It's a steep hill and looooong.

Sounds dangerous. Use a weed wacker, or just let it overgrow to tall weeds.

Southern RobotX has the SRX22T hybrid remote control lawn mower with slope mowing capabilities of up to 70 degrees. Lawn care professionals used to mow steep hills with 6 or 8 men equipped with weed eaters. Now, they are able to increase productivity, while reducing fuel consumption by 40%.

Here is a link that might be useful: Southern RobotX

I would continue to hire it out unless you plan on shelling out a lot of money. That doesn't sound like a challenge, it sounds like a death wish. Live to mow another day.

a goat?

That sounds like a pretty large area, maybe two goats.

I am not an expert and certainly not familiar with your engine, but wouldn't the oil flow out of the pan by cutting on that steep of angle?