Doom is still under copyright protection last I knew. The source is GPL, but have the assets ever been liberally licensed? I think they're more abandonware.
I'm sure you could still do it, but personally I try to respect copyright strictly for any projects I'm going to share. It just feels annoying to have copyright nonsense hanging over me otherwise.
This made me curious, so I looked at the original Doom shareware distributions on archive.org. They do include a license that allows free distribution but prohibits commercial use and generally seems to want you to not do anything other than run the software as designed. Although there are several different versions of the license and I didn't look through all of them, it's possible that some distributions were made with less restrictive licenses.
This surprised me because I thought that id's original shareware releases actually had more permissive licenses than that. Maybe the original Commander Keen did.
I guess maybe id/ZeniMax/Microsoft could theoretically sue you. But in practice the shareware assets are used completely freely without issue all over the internet.
I'm sure you could still do it, but personally I try to respect copyright strictly for any projects I'm going to share. It just feels annoying to have copyright nonsense hanging over me otherwise.