TOPIC:

I just installed a new seal in my floor jack, but have no jack oil to refill it.
On the shelf I have ATF, 10 weight fork oil, air compresser oil, power washer pump oil.
Do I really need to go out and get another oil?

I'm not an oil expert, but I'd say yes, buy some jack oil.

Regarding your thread title, it gave me a laugh.  In addition to the oils you have I've got jack oil, chain bar oil, diesel engine oil, several types, car oil, several types, lathe way oil, cutting fluid oil, thread cutting oil, bicycle chain oil, motorcycle chain oil, PB Blaster, WD-40, and don't get me started on greases... 

ATF oil is next to use in a hydraulic high pressure system.

Auto parts stores have jack oil.

A couple of years ago there was an auto parts chain in my area discontinuing Castrol full synthetic motorcycle oil. They had it priced at $1.79 per quart. Needless to say I went to every store within a ten mile range and wiped out the shelves. I got 60.

How much oil does one man need?

More.

I used to work as GSE mechanic. I work from small golf carts to big forklift.

Some hydraulic oil is thin as ATF, some are heavy weight at around 67 - 95. But as for the jack, mine looks and feel like one of the heavy weighted ones. It's definitely thicker than motor oil I've seen. Yeah, mine have that dreaded small leak from the seal as well. I have one el-cheapo China jack that is leaking and Craftman. Craftman is not leaking so I have no idea what it uses.

As for the oil, we had container full of oils. Some for gas engines, some were for big diesels, and some were for jet engines. Also had it in several 55 Gal drums that we needed to pump into container before we use it (we kept the 5qt motor oil bottles for this reason).  Will it be enough?

My jack is a Made in China Craftman. Who knows what oil they put in it. The oil that leaked out is much thicker than ATF. 

I have a floor jack given to me and it is from Harbor Freight way back when they first came to the US. It was seeping oil from the seal. I put a couple of drops of brake fluid in and topped off with jack oil. That was 20 years ago and it's still working great. The brake fluid swells the seal rubber a bit and stops the leaking. I learned that trick from an old automatic transmission mechanic. He did it if the only issue was leaking ATF instead of pulling the transmission.