On the rent vs own decision, I would like to offer my opinion (spoiler:buy).
A pancake air compressor bundle and perfectly functional flooring nailer can be had new for under 400 total. Used would likely be less.
We installed 5 in hardwood through our master bedroom (and closets), living room, kitchen, and dining room (it is probably a slight bit more than you are talking about, but I have don't have those measurements handy) some time back; there were 3 of us (myself, my wife, and my FIL) and it took us the better part of 3 weekends to complete. Granted we started around 9:30 AM, ended around 5 PM, and had a sit down lunch every day.
Still it kept three of us busy the whole time. One person would open boxes, sort the board by length, layout the flooring (in terms of patterns in the boards and joint spacing), and mark/cut those boards that needed to be trimmed; the second would fit the boards into in the flooring; and the third would come along behind the second and nail the flooring in. If the second or third person got too far ahead or caught up they would help the first. (note: we redid all the trim work at a later date so this time does not include installing shoe molding).
All in all it took 3 of us 6 days over three weekends. Home depot rents the kind of air compressor we used for $23/day, a flooring nailer for $39/day, and a nail gun for face nailing at $19/day. That would have been a total of (23+39+19)*6= 486. Of course if you can do it quicker or at the cheaper weekly rate your cost will be less.
For us buying the tools was actually cheaper than renting them. Plus it eliminated 6 trips to the hardware store (to pick up and return the tools every weekend). I also find the portable air compressor to be a very handy tool for a variety of reasons. We also anticipate removing additional carpet and installing hardwood again in the future.
TLDR: For us in the way we work and our future plans buying tools made sense over renting.