Is it possible to use a network drive as a scratch disk? I have very little space left on my only internal drive and don't want to have to purchase a drive just to connect to my PC for a couple projects.
You don't have enough disk space to do your work, and they won't let you buy more?
You might want to remind them that disk storage is cheaper than dirt.
I don't think Photoshop even lets you select a network drive for scratch, does it?
As others have pointed out, this is a bad idea, especially with large files. But you know that, so I won't lecture you (just had to say that as a disclaimer).
If you're on Windoze, which you probably are, then you should try and map a network drive on your PC and then see if you can set it as a scratch disk. A mapped drive should work, but I've not tried it. I've used mapped drives though in cases where software insisted on a local drive.
I just tried this and network drives do not appear in the scratch volumes list. This is on a Mac, but based on what I've read here in the past it's unlikely that Adobe would enable network drives as scratch drives in Windows.
If these are the constraints...
...the most workable and fastest-performing suggestion might be this: Mount your network drive and move maybe 150GB of infrequently used folders to your space on the network drive, so that the freed-up space on your internal drive can be used for your Photoshop scratch file. And if you get a low disk space error, move another 25GB of folders to the network drive; keep doing that until the errors stop.
Of course you can use network drives, they just need to be mounted as virtual drives via tunneling drivers or like e.g. Fibrechannel cards be anchored via specific BIOSes and drivers so they appear as local drives. That being said, I wouldn't try this on a normal Gigabit network. as a minor, you would need dual Gigabit or a 10GBit network.
I have never tried to use any of my networked drives for PS Scratch Disks, but have tried with various flavors of externals, and while they worked, my benchmarks showed that USB 2.0 and FW-400 (IEEE-1394a) to be horribly slow. It was not until I tested a FW-800 (IEEE-1394b), that the slowdown was minimized to the point that things worked OK. There was no USB 3.0, or eSATA, when I did those tests, so I had no benchmarks for those connections. In the case of my tests, I was using a laptop with a single SATA HDD, which was partially full. Directing my Scratch disk to that one internal (with the OS, programs, Images, etc.), was still about twice as fast as the USB 2.0, and 1.5x as fast as the FW-400. With the FW-800, it was about 1:1, with only fractions of a sec. different for the benchmarks.
Not exactly PS, but I tried to edit to/from my NAS in Premiere Pro, and while it worked, it was far to slow to allow me to actually use that setup. I did not do a benchmark, but just tried, and made observations, and quickly ruled that out, as a possibility.
Just some observations, though they do not apply to the exact question posed by the OP.