In PCI, BAR is Base Address Register, which is a register in the PCI device's configuration space which defines where in the machine's physical memory address space that particular window of memory and/or I/O will be mapped (a single device can have several BARs, for instance a simple graphics card could have one for its control registers and one for the framebuffer). So the "BAR space" would be a shorthand for "the region of the physical memory address space which can be used to map the PCI devices memory through their Base Address Registers". The size of this region is limited, and graphics cards in particular tend to have somewhat large BARs.

(See for yourself in your machine: run "lspci -v", the lines starting with "Memory at ..." or "I/O ports at ..." are the BARs.)