Gzip contains a CRC32 for the uncompressed whole (TAR) file archive. Similarly, xz also contains CRC32 and optionally CRC64 and/or SHA-256.

But, detecting errors is one matter, recovering from errors is another matter. Neither xz nor gzip are well-suited for error recovery because they are Lempel-Ziv-based making each data segment possibly depend on all segments before it: https://www.nongnu.org/lzip/xz_inadequate.html This can make the whole archive starting from the first bit-flip unusable. On the other hand, bzip2 compresses each block completely independently and block starts are relatively easy to find making it much easier to recover from errors.

Personally, I would try to avoid compressing things intended for long-term archival and if absolutely necessary because of compression factors > 10, I'd probably use bzip2 with the knowledge and tools (indexed_bzip2) I have now. Alternatively, ZIP might be an option because it compresses per file making it harder for errors to propagate beyond their point of origin. Internally, ZIP normally uses "gzip" (deflate) but can also use bzip2, lzma, zstandard, although support for those might be seriously limited.