What archive compression makes .mp4 files the smallest?
Is there an archive-based compression format that is known to make the smallest .mp4 videos?
I did a test 1080p 0:12 duration video with the only archival formats I know about:
(python3) Joshs-MBP:testing_movie mu$ ls -lS
total 12712
-rw-r--r-- 1 mu staff 2145528 Jun 6 09:26 testing.mov
-rw-r--r-- 1 mu staff 1790044 Jun 6 09:26 testing.mov.zip
-rw-r--r--@ 1 mu staff 1789512 Jun 6 09:25 testing.mov.gz
-rw-r--r-- 1 mu staff 775138 Jun 6 09:26 testing.mov.bz2
Looks like bzip2 is the best. Is there anything else that is better in terms of making the files smaller? It's ok if it takes a lot longer.
Also, I noticed you can't bzip2 a directory.
First, a minor aside about terminology: ZIP is the only archive format you used. Gzip and Bzip2 are compression formats, not archive formats. To be a bit more specific:
An archive format aggregates multiple files and/or directories, usually including metadata such as ownership, timestamps, and possibly other data, into a single file. Tar is an example of a pure archive format, it does no inherent compression,
A compresion format just compresses data, but does not inherently combine multiple files into one. Gzip, Bzip2, Brotli, LZ4, LZOP, XZ, PAQ, and Zstandard are all compression formats. Some of them (such as Gzip and LZ4) may support compressing multiple files and concatenating them into one file that can then be uncompressed into the multiple original files (which is what's happening when you gzip a directory), but they don't' store paths or other metadata, so they are not archive formats.
Some formats, such as ZIP, 7z, or RAR, combine archiving and compression (although ZIP can store files uncompressed as well).
Now, with that out of the way, let's move on to your main question:
The comment by music2myear is correct. Your results will vary widely depending on the exact specifics of the MP4 encoding used. This is because MP4 itself includes data compression, in this case optimized for compressing audio and video data without significantly lowering the perceived quality. The processes it uses for this are actually somewhat complicated (too complicated to explain here), but because of the constraint that it doesn't lower perceived quality, combined with the fact that it compresses frame-by-frame instead of as a single long stream, there's sometimes significant room for improvement (as you can see from your test).
Now, while I can't give a conclusive answer without lots of further details, I can however give you some general advice on file compression:
ZIP and Gzip show very similar results in this case because they use variants of the same compression algorithm, more specifically a derivative of the LZW algorithm known as DEFLATE. DEFLATE is not a particularly great compression algorithm, but it's ubiquitous (there are even hardware implementations of it), so it's often used as a standard of comparison. Outside of usage as a component of other file formats (such as ZIP), it's not very widely used for storage anymore. Pretty much anything based on DEFLATE (or LZW in general) isn't going to win in any respect when comparing compression algorithms.
Bzip2 by contrast does some complex transformations on the data to make it compress more efficiently, and then uses Huffman coding for the actual compression. In most cases, it compresses better than DEFLATE-based compressors, but it's slower than DEFLATE too. Because of some assumptions made in how it transforms the input data prior to Huffman coding, it is also somewhat more sensitive than many other options to how the input data is structured.
XZ uses a different algorithm called LZMA. Like the LZW algorithm that DEFLATE is derived from, LZMA is ultimately derived from an algorithm known as LZ77, though it gets insanely better compression ratios than DEFLATE-based options, and significantly better ratios than Bzip2 in most cases. In addition to LZMA, it does some transformations that make it a bit better at compressing executables than the other options. The cost of this however, is that it takes a long time to compress data. 7zip also uses LZMA, but without the data transformations, so it's often not quite as good in terms of ratios as XZ.
LZOP uses the LZO algorithm, and generally compresses worse than DEFLATE, but works much faster. Just like Gzip, it's not widely used anymore, as people tend to favor alternatives that either give a better compression ratio, or better performance.
LZ4 is a newer standard developed by Google that runs insanely fast (near memory bandwidth speed for decompression), but gets even worse compression ratios than LZO. It's been slowly supplanting LZO, as most of the stuff that used LZO used it for speed.
Brotli is another new one from Google. It's part of the HTTP/2 standard, and is specifically optimized for streaming, and can actually get both better compression ratios and performance than DEFLATE-based options do. However, it's not widely supported for plain file compression, so it may not be a viable option for your usage.
PAQ is for those who are really insanely worried about maximizing compression ratios. It uses complex combination of statistical models to achieve absolutely bonkers compression ratios (depending on the original data, it's not unusual for a file compressed with PAQ to be less than 1/10 of the original size, whereas DEFLATE averages closer to 1/2). The cost of this is, of course, that it takes an insanely long time to compress anything with PAQ. With the high compression settings, it would likely take at least half an hour to compress that sample video with PAQ. Because of the amount of time it takes, almost nobody uses PAQ, and the few who do rarely use it for anything other than archival purposes (that is, they only use it on files they are not likely to ever change).
Zstandard is the newest of the lot, and was developed by Facebook. It uses a mix of older methods and newer ones (including some machine learning techniques) to achieve compression ratios comparable to or better than bzip2 (and sometimes even better than XZ), while running significantly faster than most of the other ones I've listed except LZ4. It probably wouldn't beat XZ for your usage (and definitely won't beat PAQ), but it may get good enough ratios that the significantly better performance is worth it.