does anybody know if I can purchase a V2 5big and insert these 500 GB drives into it and recover my data?
No, that won't work. The nas boots from it's harddisks, and a V2 cannot boot from V1 disks.
If that won't work, would purchasing a used V1 5big allow me to recover my data? Does anybody know how I might find a new motherboard so that I can recover the data? If none of that is possible, then what options would I have to recover the data? I am thinking I would need to get my hands on a 5 drive SATA controller, a linux box, and some tinkering?
I'm not sure about the 2nd V1 option. The box has some nasty eeprom, which contains the UUIDs of the raid arrays, and it won't accept a raid array with a wrong UUID. I guess there should be an easy way around that, but I don't know it.
To recover your data you need to connect your disks to a Linux box. That can be anything, but a PC is the most convenient. You don't need 5 sata connections. When you have RAID5, you only need to connect 4 disks. And the disks can be connected using an USB to SATA convertor (unless they are bigger than 2TB). You can mix USB and (e)SATA if you want.