Use NICs with SFP+ slots and some twinaux SFP+ modules. The whole thing will set you back maybe about $200 for two machines and will give you 10Gbps up to 10 meters (32.4ft).
Or you can use Windows 8+ since it has SMB Multichannel and it'll boost your bandwidth between any other machine with Windows 8+ and multiple ports. I get 2Gbps between my desktop and server both running Win 10 using a pair of single port Intel NICs in each. Cost me about $60 for the two machines. A quad port Intel NIC for each would run maybe $200 for both machines for 4Gbps. Slower than a twinaux setup, but the machines are also about 15 meters distance from each other, so not viable.
No monkeying with needing 10GbE switches then, or having to have a separate 1GbE port to connect to the rest of the network.
Out of curiosity, why do you need 10GbE for back-ups? I know some businesses do, but it sounds like you need some mighty fast back-ups. Back-ups are generally done off hours or even during regular hours, shouldn't be causing a significant performance bottleneck on the machines/network. My back-ups from my desktop to my server are done at night and weekly and at worst with a 2Gbps connection it takes all of about 3-4 minutes on my bigger back-ups (about a 60GB backup). I mean, if you are actually copying everything 10GbE might be awesome, but if you are versioning or doing delta back-ups, they shouldn't be that huge (even if in the 10s of GBs, that is only a few minutes).