The source for the Linux kernel v0.01 can easily be found online, however it does not seem to contain a license or any redistribution rules. Is it, therefore, safe to assume that this code is in the public domain? Or have I missed something, and in fact it is licensed under some kind of license?
As Mark Plotnick notes, version 0.01 of Linux was released under its own, fairly liberal crayon license. The only problem with it is that it prohibits any distribution fee, which would make it GPL-incompatible. Here's the full text:
This kernel is (C) 1991 Linus Torvalds, but all or part of it may be redistributed provided you do the following: - Full source must be available (and free), if not with the distribution then at least on asking for it. - Copyright notices must be intact. (In fact, if you distribute only parts of it you may have to add copyrights, as there aren't (C)'s in all files.) Small partial excerpts may be copied without bothering with copyrights. - You may not distibute this for a fee, not even "handling" costs.
Later, in version 0.12, he chose to use GPLv2 instead. You can read a good overview of this licensing history on Wikipedia.
And just to clarify:
Is it, therefore, safe to assume that this code is in the public domain?
No. In the absence of any license, the default is all rights reserved.