Is it possible to continue copying files while switching users?

My Mac has crashed and my backup storage drive died months ago. My only hope of saving my data is by opening it in target mode on my spouse's computer. The files I am copying are extensive (8hrs-10hrs est download time), and I am wondering if it will be possible to continue copying data to his desktop and an external hard drive at the same time while allowing him to switch users so he can work simultaneously. Please help!