After some trouble with graphics drivers that caused my computer to rapidly power cycle the USB ports, my external hard drive (a Seagate One Touch 2TB) no longer appears when I plug it in to my computer. It does, however, make an intermittent beeping sound (that I initially thought was from my crashed computer), which cursory googling says has to do with the board not receiving enough power to function. The board's power light turns on like normal, but it doesn't sound like it spins up.
Using lsusb I was able to find that my computer can read the device's USB information - I can get a make and model, serial number, and some stats around its maximum current that all seem pretty much nominal - so the ROM chip is probably not fried, at least.
Outside of taking it to a data recovery shop (which, according to online estimates, I can't really afford right now), is there any possibility of recovering the data off this drive from home?