Booted up a VM into the SCCM OSD WinPE, entered password, and selected a task sequence. Before any packages were downloaded, I was hit with error 8007000E. The smsts.log file reports this error, and using the Trace Log Tool's error lookup that code means:
Not enough storage is available to complete this operation.
The VM's hard disk is IDE and 60GB Dynamic. I've never had this problem before.
The word storage in this case was a misnomer, or a red herring at best.
In this instance, the VM's RAM was the culprit. It was set to dynamic, 512MB minimum, 4GB max. The new WinPE wims are rather large, so I bumped up the minimum RAM to 2048MB, and that did the trick.
I don't know why WDS/WinPE doesn't request more RAM, or actually why it doesn't detect the max like a full OS does, whether it's a bug or by design.