I'm running unrooted stock, and Google suggested that I try a full wipe.
Since I couldn't see how this would help fix the web Play Store, I felt fairly comfortable ignoring this advice.
Some additional information I noticed: the Play Store app on my device is stuck on version v3.4.7, even though everyone else I know has gotten an OTA update to v3.5.15. I tried downloading the v3.5.15 APK and installing it manually, and this resulted in me the "Error retrieving information from server. [RPC]" message that was described on AndroidCentral.
Incidentally, it was interesting to see how many people on the comments section there were having the same problem. I suspect this problem might actually be more widespread, and once the mainstream tech sites start to pick up on it we might start to get a better idea of what the real extent is.